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THE 

PHYSICAL SOCIETY 
LONDON. 



REPORT 



OK THE 



RELATIVITY THEOEY OF 
GRAVITATION. 



A. S. EDDINOTON, M.A., M.Sc., F.E.S. 

M 

Plumtan Professor of Astronomy and Experimental Philosophy, Cambridge. 



Price to Non-Fellows, 6 8. net, post free 6s. 3d, 
Bound in doth, 8s. 6d., post /ree 8s 9d 



LONDON. 

FLEET WAY PRESS, LTX , 
1, 2 AND 3, SALISBURY COUBT, FLEET STBEET 

1920. 



CONTENTS. 



CHAPTEE L 

PAGE 

THE KESTRICTED PRINCIPLE or EELATIVITY 1 

1-3. The Michelson-Morley experiment and its significance. 

4. The transformation of co-ordinates for a moving observer. 

5. Reciprocity of the transformation. 6. Standpoint ot the 
Principle of Relativity. 7, Transformation of velocity, of den- 
sity and of mass. 8. Scope of the Principle. 



CHAPTER II. 
THE KELATIONS OF SPACE, TIME, AND FORCE 14 

9-10. Minkowski's transformation. 11. Invanance of 8$. 12. 
Irrelevance of co-ordinate systems to the phenomena, 13-14. 
The Principle of Equivalence. 15-16. Definition of a field of 
force by g^ v . 17. Purpose of the theory of tensors. 18, Nature 
of space and time in the gravitational field. 



CHAPTEE III. 

THE THEORY OF TENSORS 30 

19. Notation , definition and elementary properties of tensors. 

20. The fundamental tensors ; associated tensors. 21. Auxili- 
ary formula for the second derivatives of the co-ordinates. 22. 
Covariant differentiation. 23-24. The Riemann-Ghristoffel ten- 
sor. 26. Summary. 

CHAPTEE IV. 
EINSTEIN'S LAW OF GRAVITATION 41 

26. The contracted Eiemann-Christoffel tensor. 27. Limitation 
of the Principle of Equivalence. 28. The gravitational field of a 
Dartiole* 



IV. CONTENTS. 

CHAPTER V. 

PAGB 

THE CRUCIAL PHENOMENA 48 

29-30. The Equations of Motion. 31. Motion of the Perihelion 
of Mercury. 32-33. Deflection of a ray of hght 34. Displace- 
ment of spectra] lines. 

CHAPTER VL 

THE GRAVITATION OF A CONTINUOUS DISTRIBUTION OF 

MATTER 59 

35-36 Equations for a continuous medium. 37 The energy- 
tensor T v , and the equations of hydrodynamics. 38. The Law 
of Conservation 39. Reaction of the gravitational field on 
matter. 40. Propagation of gravitation 

CHAPTER VII. 
THE PRINCIPLE OF LEAST ACTION 71 

41. Expression ot the law of gravitation in the torm ot Lagrange's 
Equations 42. Principle of Least Action. 43. Energy of the 
gravitational field 44 Method of Hilbert and Lorentz. 45-46. 
Electromagnetic equations. 47 The JEJther. 48. Summary of 
the last two chapters 

CHAPTER VIII. 

THE CURVATURE OF SPACE AND TIME 82 

49. Absolute rotation and the limits ol generalised relativity. 
50 Einstein's curved space 51. T)e Sitter's curved space-time. 
52. Boundary Conditions. 53. Conclusion 



PREFACE TO FIRST EDITION. 



THE relativity theory of gravitation in its complete 
form was published by Einstein in November 1915. 
Whether the theory ultimately proves to be correct 
or not, it claims attention as one of the most 
beautiful examples of the power of general mathematical 
reasoning. The nearest parallel to it is found in the applications 
of the second law of thermo-dynamics, in which remarkable 
conclusions are deduced from a single principle without any 
inquiry into the mechanism of the phenomena ; similarly, if 
the principle of equivalence is accepted, it is possible to stride 
over the difficulties due to ignorance of the nature of gravita- 
tion and arrive directly at physical results. Einstein's theory 
has been successful in explaining the celebrated astronomical 
discordance of the motion of the perihelion of Mercury, 
without introducing any arbitrary constant ; there is no trace 
of forced agreement about this prediction. It further leads to 
interesting conclusions with regard to the deflection of light 
by a gravitational field, and the displacement of spectral lines 
on the sun, which may be tested by experiment. 

The arrangement of this Report is guided by the object of 
reaching the theory of these crucial phenomena as directly as 
possible. To make the treatment rather more elementarv, 
use of the principle of least action and Hamiltonian methods 
has been avoided ; and the brief account of these m Chapter 
VII. is merely added for completeness. Similarly, the equa- 
tions of electro-dynamics are not used in the main part of 
the Report. Owing to the historical tradition, there is an 
undue tendency to connect the principle of relativity with 
the electrical theory of light and matter, and it seems well to 
emphasize its independence. The main difficulty of this 
subject is that it requires a special mathematical calculus, 
which, though not difficult to understand, needs time and 
practice to use with facility. In the older theory of relativity 
the somewhat forbidding vector products and vector operators 



VI. PEEFACE TO FIRST EDITION. 

constantly appear. Happily this can now be avoided alto- 
gether ; bat in its place we use the absolute differential 
calculus of Kicci and Levi-Civita. This is developed ab imtio 
so far as required in Chapter III. Attention must be called 
to the remark on notation in 19 a which concerns almost all 
the subsequent formulae. 

Extensive use has been made of the following Papers, which 
in some places have been f otlowed rather closely : 

A. EINSTEIN. Die Gmndlage der allgemeinen Relitivitats 
theorie " Annalen der Physik, 57 XLIX., p. 769 (1916). 

W. DE SITTER. On Einstein's Theory of Gravitation and 
its Astronomical Consequences. " Monthly Notices of the 
Royal Astr. Soc.," LXXVL, p. 699 (1916) ; LXXV1L, p, 155 
(1916) ; LXXVHL, p. 3 (1917). 

I am especially indebted to Prof, de Sitter, who has kindly 
read the proof-sheets of this Report. 

The principal deviations in the present treatment of the 
subject will be found in Chapter VI. I have ventured to 
modify the enunciation of the principle of equivalence in 27 
in order to give a precise criterion for the cases in which it is 
assumed to apply. 

Other important Papers on the subject, most of which have 
been drawn on to some extent, are . 

EL HILBERT. Die Grundlagen der Physik, Cf Gottingen 
Nachrichten," 1915, Nov. 20. 

EL A. LOREOTZ. On Einstein's Theory of Gravitation, 
1 Proc. Amsterdam Acad.," XIX., p. 1341 (1917), 

J. DROSTE. The Field of n moving centres on Einstein's 
Theory, " Proc. Amsterdam Acad.," XIX., p. 447 (1916). 

A. EINSTEIN. Kosmologische Betrachtungen zur allge- 
meinen Relitivitatstheorie, " BeilinSitzungsber.," 1917, Feb. 8. 
Ueber Gravitations wellen, ibid, 1918, Feb. 14. 

K, SCHWARZSCHILD. Ueber das Gravitationsfeld eines 
Massenpunktes nach der Einstein'schen Theorie, " Berlin 
Sitzungsber,/' 1916, Feb. 3. 

T. LEVI-CIVITA, Statica Einsteiniana, " Rendiconti dei 
Lincei," 1917, p, 458. 

A, PALATINI. Lo Spostamento del Perielio di Mercuric 
" Nuovo Cimento," 1917, July. 

The last two Papers avoid much of the heavy algebra, but 
claim a rather extensive knowledge of differential geometry, 

The older theory of relativity, briefiy surveyed in the first 
chapter, is fully treated in the well-known text-books of L. 



PREFACE TO FIRST EDITION. VU, 

Silberstein (Macmillan & Co.) and E. Cunningham (Canib. 
Univ. Press) A useful review of the mathematical theory of 
Chapter III., giving a fuller account from the standpoint of 
the pure mathematician, will be found in " Cambridge Mathe- 
matical Tracts, 55 No. 9, by J. E. Wright. Finally, for those 
who wish to learn more of the outstanding discrepancies 
between astronomical observation and gravitational theory, 
the following references may be given : 

W. DE SITTER. The Secular Variations of the Elements of 
the Four Inner Planets, " Observatory," XXXVI., p. 296. 

E. W. BROWN. The Problems of the Moon's Motion, " Ob- 
servatory," XXXVII., p. 206. 

H. GLAUERT. The Rotation of the Earth, " Monthly Notices 
of the Eoyal Astr. Soc.," LXXV., p. 489. 



PREFACE TO SECOND EDITION. 



THE advances made in the eighteen months since 
this Report was written do not seem to call for 
any modification in the general treatment. Perhaps 
the most notable event is the verification of Ein- 
stein's prediction as to the deflection of a ray of light by 
the sun's gravitational field. This was tested at the total 
eclipse of May 29, 1919, at two stations independently, by 
expeditions sent out by the Royal and Royal Astrono- 
mical Societies jointly, under the superintendence of the 
Astronomer Royal. The deflection, reduced to the sun's 
limb should be l"-75 on the relativity theory, and 0"-87 (or 
possibly zero) according to previous theories. At Principe, 
where the observations were very much interfered with by 
cloud, the value 1"-61 was obtained, with a probable error 
of 0"-3 ; the accuracy appears to be sufficient to indicate 
fairly decisively Einstein's value. At Sobral, where a clear 
sky prevailed, the observed value was 1"*98 ; the accordance 
of results derived from right ascensions and declinations, 
respectively, and the agreement of the displacements of 
individual stars with the theoretical law demonstrate in a 
particularly satisfactory manner the trustworthiness of the 
observations at this station. The full results will be pub- 
lished in a Paper by Sir F. W. Dyson, A. S. Eddington, and 
C. Davidson in the Philosophical Transactions of the Royal 
Society. 

The test of the displacement of the Fraunhofer lines to the 
red stands where it did, and we still think that judgment must 
be reserved. In view of the possibility of a failure in this 
test, it is of interest to consider exactly what part of the 
theory can now be considered to rest on a definitely experi- 
mental basis. I think it may now be stated that Einstein's 
law of gravitation is definitely established by observation in 
the following form : 



X. PREFACE TO SECOND EDITION. 

Every particle and light-pulse moves so that the integral 
of ds between two points on its track is stationary, where 
(equation (28-8)) 

&= (1 2m/r) - l dr* rW r 2 sin 2 6d<p* + (I2mfr)dt* 
in appropriate polar co-ordinates, the co-efficient of dr* being 
verified to the order m/r, and the co-efficient of dt* to the 
order m 2 /r 3 . This is checked for high speeds by the deflection 
of light, and for comparatively low speeds by the motion of 
perihelion of Mercury, so that unless the true law is of a kind 
much more complicated than we have allowed for, our ex- 
pression cannot well be in error. 

Accepting Einstein's law in this form, the properties of 
invariance for transformations of co-ordinates follow, and we 
reach the conclusion that the intermediary quantity ds (to 
which as yet we have assigned no physical interpretation) is 
an invariant, that is to say it has some absolute significance 
in external nature. 

Einstein's theory (as distinct from his law of gravitation) 
gives a physical interpretation to ds, as a quantity that can 
be measured with material scales and clocks. It is this 
interpretation which the observation of the Fraunhofer lines 
should test. The quantity ds is an ideal measure of space and 
time ; and it is possible that we have not yet reached finality 
as to the right way of realising the ideal practically. It is a 
fair prediction that an atomic vibration will register ds like 
an ideal clock ; and it is difficult to seeliow this can be avoided 
unless the equations of vibration of an atom involve the 
Briemann-Christofiel tensor. But, if the test fails, the logical 
conclusion would seem to be that we know less about the 
conditions of atomic vibration than we thought we did. 

A very notable extension of the theory to include electro- 
magnetic forces and gravitational forces in one geometrical 
scheme has been given by Prof. H. Weyl in two Papers 

Berlin, Sitzungsberichte, 1918, May 30. 
Annalen der Physik, Bd. 59, p. 101. 

In Einstein's theory it is assumed that the interval ds has 
an absolute value, so that two intervals at diSerent points 
of the world can be immediately compared. In practice 
the comparison must take place by steps along an intermediate 
path ; for example, by moving a material measuring rod 
from one point to the other continuously along some path. 
It is possible that the result of the comparison may not be 



PREFACE TO SECOND EDITION. 33- 

independent of the path followed, and Weyl considers the 
electromagnetic field to be the manifestation of this incon- 
sistency. This leads to a very beautiful generalized geometry 
of the world, in which the electromagnetic field appears as 
the sign of non-integrability of gauge, and the gravitational 
field as the sign of non-integrability of direction. The theory 
has important consequences though it has not suggested 
any experimental test. It may be added that it appears 
to favour Einstein's view of the curvature of space, which 
has been treated, perhaps too unsympatlietically, m Chapter 
VIII. 

The writer holds the view that the fundamental equations 
of gravitation (35-8), which on this theory are the sole basis 
of mechanics, should be regarded as a definition of matter 
rather than as a law of nature. We need not suppose that 
the gravitational field has in vacuo some innate tendency to 
arrange itself according to the law G>=0 ; we should rather 
say that in regions of the world where this state happens to 
exist we perceive emptiness ; and where the equations fail, 
tke failure of the equations is itself the cause of our perception 
of matter. Matter does not cause the curvature (&) of space- 
time ; it is the curvature. Just as light does not cause electro- 
magnetic oscillations ; it is the oscillations. This point of 
view is developed in a Paper which will appear shortly ia 
" Mind. 55 

Finally, a word may be added for those who find a difficulty 
in the combination, of space and time into a static four- 
dimensional world, in which events do not " happen " they 
are just there, and we come across them successively in our 
exploration* " Surely there is a difference between the 
irrevocable past and the open future, different in quality 
from the arbitrary distinction of right and left." We agree 
entirely ; but this difference, whatever it is, does not enter 
into the determinate equations of physics. For physics, the 
future is + t and the past 1 } just as right is + a? and left x. 
If we change the place of one particle in our problem we alter 
the past as well as the future, in contrast to what appears to 
be the ordinary experience of life, that our interference will 
alter the future but not the past. The static four-dimensional 
representation may thus be not completely adequate, but it 
suffices for all that comes within the purview of physics. 

December, 1919. 

B2 



CHAPTEB I. 



THE BESTBICTED PRINCIPLE OF RELATIVITY. 

1. In 1887 the famous Michelson-Morley experiment was 
performed witli the object of detecting the earth's motion 
through the sether. The principle of the esrperiment may be 
illustrated by considering a swimmer in a river.^ It is easily 
realized that it takes longer to swim to a point 50 yards 
up-stream and back than to a point 50 yards aeross-stream 
and back. If the earth is moving through the aether there is 
a river of sether flowing through the laboratory, and a wave 
of light may be compared to a swimmer travelling with 
constant velocity relative to the current. If, then, we divide a 
beam of light into two parts, and send one half swimming up 
the stream for a certain distance and then (by a mirror) back 
to the starting point, and send the other half an equal distance 
aeross-stream and back, the aeross-stream beam should arrive 
back first. 

Let the sether be flowing relative to the apparatus with 
velocity u in the direction Ox (Fig. 1) ; and let OA, OB be 
the two arms of the apparatus of equal length a 9 OA being 
placed up-stream. Let v be the velocity of light. The time 
for the double journey along OA and back is 

2 - 2 V (1-1) 

*~~ P * ' ' l ' 



where /3=(1 u*/v z )"~%, a factor greater than unity. 

Fof the transverse journey the light must have a com- 
ponent velocity u up-stream (relative to the sether) in order 
to avoid being carried below OB ; and, since its total velocity 
is ?, its component aeross-stream must be \S(v*~u z ). The 
fame for the double journey OB is accordingly 

* 2a -** 

*~V( 

so that #! > 2 . 



2 BELATIVITY THEORY OF GRAVITATION. 

But when the experiment was tried, it was found that l>oth 
parts of the beam took the same time, as tested by the inter- 
ference bands produced. It would seem that OA and OB 
could not really have been of the same length ; and if OB 
was of length a l5 OA must have been of length ajp. The 
apparatus was now rotated through 90, so that OB became 
the up-stream arm. The time for the two journeys was again 
the same, so that OB must now be the shorter arm. The plain 
meaning of the experiment is that both arms have a length a l 
when placed along Oy, and automatically contract to a length 
0J//S when placed along Ox. This explanation was first given 
by FitzGerald. 

It is not known how much the earth's motion through the 
aether amounts to ; but at some time during the year it must 



FIG. 1. 



be at least 30 km, per sec,, since the earth's velocity changes 
by 60 km. per sec. between opposite seasons. The experiment 
would have detected a velocity much smaller than this (about 
$ km. per sec.), if it were not for the compensating contraction 
of the arms of the apparatus. By experimenting at different 
times of the year with different orientations the existence of 
the contraction has been fully demonstrated. It has been 
shown that it is independent of the material used for the arms, 
and the contraction is in all cases measured by the ratio 

===(! -ttV^r*- 

It is now known that this contraction fits in well with the 
electrical theory of matter, and may be attributed to changes 
in the electromagnetic forces between the particles which 
determine the equilibrium form of a so-called rigid body. 
This umveisal property of matter is therefore not so mysterious* 



THE BESTBICTED PRINCIPLE OF BELATIVITY. 3 

as it at first seemed ; and we sliall not here discuss the un- 
successful attempts at alternative explanations of the 
Michelson-Morley experiment, e.^., by assuming a convection 
of the eether by the earth. 

2. The Michelson-Morley experiment has thus unexpectedly 
failed to measure our motion through the sether, and many 
other ingenious experiments have failed in like manner. So 
far as we can test, the earth's motion makes absolutely no 
difference in the observed phenomena ; and we shall not be 
led into any contradiction with observation if we assign to the 
earth any velocity through the aether that we please. It is 
interesting to trace in a general way how this can happen. 
Let us assign to the earth a velocity of 161,000 miles a second, 
say, in a vertical direction. With this speed /?=2, and the 
contraction is one-half. A rod 6 feet long when horizontal 
contracts to 3 feet when placed vertically. Yet we never 
notice the change. If the standard yard-measure is brought 
to measure it, the rod will still be found to measure two yards ; 
but then the yard-measure experiences the same contraction 
when placed alongside, and represents only half -a-yard in that 
position. It might be thought that we ought to see the change 
of length when the rod is rotated. But what we perceive is 
an image of the rod on the retina of the eye ; we think that 
the image occupies the same space of retina in both positions ; 
but our retina has contracted in the vertical direction without 
our knowing it, and our estimates of length in that direction 
are double what they should be. Similarly with other tests. 
We might introduce electrical and optical tests, in which the 
cause of the compensation is more difficult to trace ; but, in 
fact, they all fail The universal nature of the change makes 
it impossible to perceive any change at all. 

3. This discussion leads us to consider more carefully what 
is meant by the length of an object, and the space which we 
consider it to occupy. To the physicist, space means simply 
a scaffolding of reference, in which the mind instinctively 
locates the phenomena of nature. Our present point of view 
assumes that there is a " real " or " absolute " scaffolding, 
in which a material body moving with the earth changes its 
length according as it is oriented in one direction or another. 
On the other hand, the human race (and its predecessors) have 
conceived and used a different scaffolding the space of 
appearance in which a material body moving with the earth 
does not change length as its orientation alters. It often 



<1 RELATIVITY THEORY Off GRAVITATION. 

happens that a primitive conception is ambiguous, and lias 
to be re-defined when adopted for scientific purposes ; but 
there is little justification for doing this in the case of space. 
Firstly, the space of appearance is perfectly suitable for 
scientific purposes, since we have just seen that it is impossible 
to detect experimentally that it is not the absolute space 
Secondly, so long as we cannot detect our motion through 
the aether, we do not know how to convert our observations 
so as to express them in terms of absolute space. Thirdly, 
for all we know, our velocity through the aether may be so 
great that the absolute space and the space of appearance do 
not even approximately correspond ; thus we might be re- 
volutionising rather than re-defining the common conception 
of space. 

It will therefore be considered legitimate to use the words 
" space " and ** length " with their current significance A 
rigid body on the earth is generally considered not to change 
length when its direction is altered, and by this property we 
block out a scaffolding of reference for our measures and 
locate objects in our space the space of appearance. But 
we have learnt one important thing. Our space is not abso- 
lute ; it is determined by our motion. If we transler our- 
selves to the star Arcturus, which is moving relatively to us 
with a speed of more than 300 km. per sec., our space will 
not suit it, since it was designed to eliminate our own con- 
traction effects. The contraction ratio ft must be different 
for Arcturus ; and the space surveyed with a material yard- 
measure carried on Arcturus will differ slightly from the space 
surveyed with the same yard-measure on the earth. It may 
also be noted that there is a slight difference in our own space 
in summer and winter (owing to the change of the earth's 
motion), and this may have to be taken into account in some 
pplications. 

Accordingly by " space " we shall mean the space of appear- 
ance for the observer considered. It becomes definite when we 
specify the motion of the observer. In particular, if the 
observer is at rest in the aether, the corresponding space is 
what we have hitherto called the " absolute space." 

The possibility of different observers using different spaces 
may be illustrated by considering the question, What is a 
circle ? Suppose a circle is drawn on paper in the usual way 
with a pair of compasses. An observer 8, who believes the 
paper to be moving through the aether with a great velocity, 



THE RESTRICTED PRINCIPLE OF RELATIVITY. 5 

mast, in accordance with the Michelson-Morley experiment, 
suppose that the distance between the points of the compasses 
changed as the curve was described ; he will therefore deem 
the curve to be an ellipse. Another observer S' 3 who believes 
the paper to be at rest in the s&ther, will deem it to be a circle. 
There is no experimental means of finding out which is right 
in his hypothesis. We have, therefore, to admit that the 
same curve may be arbitrarily regarded as an ellipse or as a 
circle. That illustrates our meaning when we say that S and 
S' use different spaces, the curve being an ellipse in one space 
and a circle in the other 

4. The failure of all experimental tests to decide whether 
the space of S or of S' is the more fundamental is summed up 
in the restricted Principle of Eelativity. This asserts that 
it is impossible by any conceivable experiment to detect uniform 
motwn through the cether. This generalisation is based on a 
great amount of experimental evidence, which is fully dis- 
cussed in text-books on the older theory of relativity. Here 
it is perhaps sufficient to state that experimental confirmation 
appears to be sufficient,* except in regard to the question 
whether gravitation falls within the scope of the principle. 
We shall assume that the principle is true universally. 

Let x', y\ z', be the co-ordinates of a point in the space 
of an observer S' ; and let x, y, z be the co-ordinates of the 
same point in the space of an observer S at rest in the sether. 
Let $' move relatively to S with velocity u in the direction Ox. 
S', using his own space, has no knowledge of his motion through 
the aether, and he makes all his theoretical calculations as 
though he were at rest ; from what has been already said, 
he will not discover any contradiction with observation. 

According to ordinary kinematics the relation between the 
co-ordinates and the times (t f , t) in the two systems would be 

x'=x-ut, ?/=/, z'=z, *'= . . . (4-1) 

But the first of these must be modified, because in the in- 
direction S"s standard of length is contracted in the ratio I/ ft. 
The equation becomes * 

%'=p(x~ut). (4-15) 

In order to satisfy the principle of relativity, it appears that 
the time tf used by S' must differ from the time t used by 8. 

* /.., sufficient to assert 6he wnfo&attltly, aot necessarily the perteefc 
accuracy, of the principle. 



6 RELATIVITY THEORY OF GRAVITATION. 

We shall suppose that both observers use the same value for 
the velocity of light ; this is merely a matter of co-ordinating 
their units, the significance of which will be considered in the 
next paragraph. Let S' observe the time t' taken for the 
double journey OB=2a^ in Kg. 1. It must agree with his 
calculated time, which is, of course, Za^v. Thus 



But in (1-2), when we were using $'s co-ordinates, we found 
the time to be 

t=2aip/v. 
Hence 

*=#'. 

This also fits the double journey OA. $', unaware of his 
motion, does not allow for any contraction, and calculates the 
time for the double journey as 



But S recognises the contraction, and considers the distance 
travelled to be %a lf j j3. Hence calculating as in (1*1), he makes 
the time to be 



so that again t=j3t'. 

Accordingly S' must use a unit of time longer than that of 
S in the ratio ft ; otherwise he would find a discrepancy 
between observation and calculation. 

There is another difference in time-measurement involved. 
According to S, the light completes the half -journey OA in a 

time ~-^~ in /S's units, or in $ n s units of time. But 

v u vu 



But the difference in the time of leaving and reaching A 
must be deemed by S' to be a,i/v ; he must therefore set his 
clock at A a-iU/v* slow compared with the clock at O, He 
has no idea that it is slow ; he has attempted to adjust the 
two clocks together. But his determination of simultaneity 
of events at and A differs from that of S, because he allows 
a different correction for the time of transit of tlie light. 



THE RESTRICTED PRINCIPLE OF BELATIVITY. 7 

Including both these differences, we see that the relation 
between the times adopted by S and S' is 



Substituting this value of t in (4 15) we obtain after an easy 

reduction 

x^P(x'+ut'}. 

Collecting together our results, we have the formulae of 
transformation 



By the principle of relativity nothing is altered if 8 is in 
motion relative to the aether ; so the relations (4-2) must hold 
between the spaces and times of any two observers having 
relative velocity u. 

By solving (4-2) for a;', y f , %', t', we obtain the reciprocal 
relations 



These might have been written down immediately, because 
interchanging S and S' is equivalent to reversing the sign of u ; 
but it will be seen later that the verification by direct solution. 
of (4-2) is important. 

5. We have supposed that S and S' adopt the same measure 
for the velocity of light ; this was in order to secure^that the 
units of velocity used by S and S' correspond. It is no use 
for S to describe his experiences to S' in terms of units which 
are outside the knowledge of the latter ; but if S states that a 
velocity occurring in his experiment is a certain fraction of the 
velocity of light> S' will be able to compare that with his own 
experimental results. By the principle of relativity any other 
velocities occurring in their experiments under similar con- 
ditions will correspond ; and, for example, we see from (4 2) 
and (4*3) that they will agree in calling their relative velocity 
-\-u and u respectively. 

Whilst this settles the consistency of the units of velocity 
used in (4-2) we have not yet secured that the units of length 
correspond. A description of Brobdingnag by a Brobding- 
nagian would not have mentioned the most striking feature 
of that country ; it needed an intruding Gulliver to detect 



8 RELATIVITY THEORY OF GRAVITATION. 

the enormous scale of everything contained. And so we may 
ask whether a natural standard of length, say a hydrogen 
atom, at rest in $ 5 s system will be of the same size in terms 
of x, y, z, as a hydrogen atom at rest in 8"$ system in terms 
of x\ y\ z\ Clearly it will be misleading if we do not correlate 
the co-ordinates so as to satisfy this. 

To allow for a possible non-correspondence oi the units of 
length in (4-2) we can. write the transformation more generally 

ib^jS^'+wO, ty^y\ kz=z', H^f}(t'+ux'lv*) , (5-2) 

where Jc depends on the magnitude, but clearly not on the 
direction, of u. 

But now applying (5-2) the reverse way, i.e., regarding 
x, t/, z 9 t as a system moving with velocity u relative to 
&'? y'> %'> t', we shall have 

fo^/Jfc-ttf), %'=y, feW, Jct'=:p(t-ux/v 2 ) . (5-3) 

which is clearly inconsistent with (5-2) unless k=l. Hence (4*2) 
gives the only possible correspondence of the units of length* 

We thus use the remarkable property of reciprocity possessed 
by (4-2) and (4-3), but not by (5-2) and (5-3), to fix the necessary 
correspondence of the units. The dimensions of a motionless 
hydrogen atom will now be the same in' both systems ; for, if 
not, we could find a system in which the dimensions were 
either a maximum or a minimum ; and that system would 
give us an absolute standard from which we could measure 
absolute motion 

It is thus clear that S' will actually measure his space and 
time by the variables x\ y', z', t' given by (4-3), if he sets 
about choosing his units in the same way that S did. 

6. We have established the connection between the co- 
ordinates used by 8 and S' by reference to simple criteria. 
It is interesting to work out in detail the correspondence of 
the two systems for other and more complex phenomena, 
showing that the transformation always works consistently. 
But the standpoint of the 'principle of relativity rather dis- 
courages this procedure. Its view is that the indifference of 
all natural phenomena to an absolute translation is something 
immediately understandable, whilst the contractions and other 
complications entering into our description arise from our own 
perversity in not looking at Nature in a broad enough way. 
When a rod is started from rest into uniform motion, nothing 
whatever happens to the rod. We say that it contracts ; but 



THE RESTRICTED PRINCIPLE OF RELATIVITY. 9 

length is not a property of the rod ; it is a relation between 
the rod and the observer. Until the observer is specified the 
length of the rod is quite indeterminate. We ought always to 
remember that our experiments reveal only relations, and not 
properties inherent in individual objects ; and then the corre- 
spondence of two systems, differing only in uniform motion, 
becomes axiomatic, so that laborious mathematical verifica- 
tions are redundant. Human minds being what they are, that 
is a counsel of perfection, and we shall not follow it too strictly. 

The only verification that is needed is to show that our 
fundamental laws of mechanics and electrodynamics are con- 
sistent with the principle of relativity. This will be done in 
connection with a much more general principle of relativity 
for mechanics in 37, and for electrodynamics in 45. *~ 

7. (a) As an illustration of the modification of ordinary 
views required by this theory, we may notice the law of 
composition of velocities. Consider a particle moving relative 
to S with velocity w along Ox, so that 



The velocity relative to <S' will be 



The velocity relative to S' is thus not w u, as we should 
have assumed in ordinary mechanics. 

It has been pointed out by Robb that the addition-law for 
motion in one dimension can be restored if we measure motion 
by the rapidity, tanhr^w/u), instead of by the velocity w. 
Equation (7-2) gives 



{w/'y) . . (7-3) 

Since tanh" 1 l=oo , the velocity of light corresponds to 
infinite rapidity, and we may compound any number of 
relative velocities less than that of light without obtaining a 
resultant greater than the velocity of light, 

(&) To find the relation of the densities (o-=number of par- 
ticles per unit volume) in the two systems, we can easily verify 
that the Jacobian a(j/, y', z\ t')/d(x, y, z, *)=!, so that 

dx'dy'ikfdt'dxdydzdt ..... (7*4) 



10 RELATIVITY THEORY OF GRAVITATION. 

But the number of particles in a particular element of volume 
cannot depend on the co-ordinates used to describe the element, 
hence 

o-'dx'dy'dz'=vdxdydz. . . (7-5) 

<r r dt' ( uw\ 

Hence == :ji ==/M * r) ..... 

<r dt r \ iP i 

since dx/dt=w. 

In particular, if ^=0, so that v is the density referred to 
axes moving with the matter, 

cr'=cr ........ (7-65) 

Since the mass of a particle may depend on its motion, we 
cannot assume that the ratio p'/p of the mass-density is the 
same as that of the distribution-density <r'/cr. 

When the transformation (4-2) was first introduced in 
electro-dynamics by Larmor and Lorentz, if was regarded as 
a fictitious time introduced for mathematical purposes, and it 
was scarcely realised that it was the actual measured time of 
the moving observer. Einstein in 1905 first showed that 
velocity and density would be estimated by the moving 
observer in the way given above, and thus removed the last 
discrepancy between the electrodynamical equations for the 
two systems. 

(c) In order to find the change (if any) of mass with velocity, 
consider a body of mass m 1? m/ (in the two systems of reference) 
moving with velocity w ly w. Let 



Working out />/ by using (7-2), we easily find 

AiW=^iK^) ..... (7-71) 

Let a number of bodies be moving in a straight line subject 
to the conservation of mass and momentum, i.e., 

2m l and Zm 1 w\ arc conserved. 
Then, since u and p are constants, 

pZm^w^u) will be conserved. 
Therefore by (7-71) 

S~~^~Wi is conserved. . . . (7-72) 

But since momentum must be conserved for the observer S f 
Sm t 'wi is conserved .... (7-73) 



THE RESTRICTED PRINCIPLE OF RELATIVITY, 11 

The results (7 72) and (7-73) will agree if 

Wt m/ 
*=-5- > =w , say, 

Pi Pi 

and it is easy to show that there is no other solution. Hence 
m^moP^m^l-wf/v*)-* .... (7-8) 
where w is constant and equal to the mass at rest. This is 
the law of dependence of mass on velocity. 

Neglecting w^jv\ we have 

Wl== ^ +(i mo w 1 2 )/t; 2 .... (7-85) 
so that we may regard the mass as made up of a constant 
mass m belonging to the particle, together with a mass pro- 
portional to, and presumably belonging to, the kinetic energy. 
If we choose units so that the velocity of light is unity, the 
mass of the energy is the same as the energy, and the dis- 
tinction between energy and mass is obliterated. Accordingly 
m is also regarded as a form of energy. (It is usually identified 
mainly with the electrostatic energy of the electrons forming 
the body ) 

Since the conservation of mass now implies the conservation 
of energy we have to restrict the reactions between the bodies 
in the foregoing discussion to perfectly elastic impacts. Other 
interactions would require a more general treatment ; in fact, 
if the energy is not conserved, the momentum is not perfectly 
conserved, because the disappearing energy has mass and 
therefore carries off momentum. 

In this discussion we are justified in pressing the laws of 
conservation of mass and momentum to the utmost limit as 
holding with absolute accuracy, since the definition and 
measurement of mass (inertia) rests on these laws,* and 
unless we have an accurate definition it is meaningless to 
investigate change of mass. In astronomy, however, the 
masses of heavenly bodies are measured by their gravitational 
effects ; naturally we cannot legitimately apply (7-8) to 
gravitational mass without a full discussion of the law of 
gravitation. 

It should be noticed that this change of mass with velocity 
is in no way dependent on the electrical theory of matter. 

* The mass here discussed is sometimes called the " transverse mass." 
The so-called longitudinal mass is of no theoretical importance t it is not 
conserved, it does not enter into the expression, for th momentum or energy, 
and it has no connection with gravitation. 



12 KELATIVITY THEORY OF GRAVITATION. 

(d) To find the transformation of mass-density, p, we have 

which becomes by (7-71) 

(7-91) 



In particular, if p is the density in natural measure, i.e., 
referred to axes moving with the matter, p the density referred 
to axes with respect to which the matter has a velocity u, 

p=/8 a Po (7-92) 

8. Of late years the domain of the electromagnetic theory 
has been extended, so that most natural phenomena are now 
attributed to electrical actions. The relativity theory does 
not presuppose an electromagnetic theory either of matter or 
of light ; but, if we accept the latter theories, it becomes 
possible to state exactly the points on which experimental 
evidence is required in order to establish our hypothesis. 
The experimental laws of electromagnetism are summed up 
in Maxwell's equations ; and in so far as these cover the 
phenomena, the complete equivalence of the sequence of events 
in a fixed system described in terms of x, y, z, t, and a moving 
system described in terms of x\ y' y z' 3 ', has been established 
analytically. So far as is known, only three kinds of force are 
outside the scope of Maxwell's equations. 

(1) The forces which constrain the size and shape of an 
electron are not recognised electromagnetic forces. For- 
tunately the properties of an electron at rest and in ex- 
tremely rapid motion can be studied experimentally, and 
it is believed that they change in the way required by 
relativity. 

(2) The phenomena of Quanta appear to obey laws outside 
the scope of Maxwell's equations. Theoretically these laws 
fit in admirably with relativity, since Planck's fundamental 
tuiit of action is found to be unaltered by the choice of axes. 
But on the experimental side, evidence of the relativity of 
phenomena involving quantum relations has not yet been 
produced. This is particularly unfortunate, because the 
vibration of an atom depends on quantum relations ; and it is 
practically essential to the relativity theory that an atom 
(acting as a natural clock) should keep the time appropriate 
to the axes chosen 



THE RESTRICTED PRINCIPLE OF RELATIVITY. lb 

(3) Gravitation is outside tlie electromagnetic scheme. The 
Michelson-Morley experiment is necessarily confined to solids 
of laboratory dimensions, in which internal gravitation has 
no appreciable influence. There is, therefore, no experimental 
proof that a body such as the earth, whose figure is determined 
mainly by gravitation, will undergo the theoretical contraction 
owing to motion. The most direct evidence that gravitation 
conforms to relativity comes from a discussion by Lodge* of 
the effect of the sun's motion through the sether on the peri- 
helia and eccentricities of the inner planets. If gravitation is 
outside the relativity theory (the Newtonian law holding 
unmodified) a solar motion of 10 km. per sec. would produce 
perturbations in the eccentricities and perihelia of the earth 
and Venus, which could probably be detected by observation. 
The absence of these perturbations seems to show that gravita- 
tion must conform to relativity, unless, indeed, the sun happens 
to be nearly at rest in the sether. If we confine attention to 
our local stellar system the average stellar velocities are not 
so much greater than 10 km. per sec. as to render the lattei 4 
alternative too improbable ; but the very high velocities found 
for the spiral nebulae (which are thought to be distant stellar 
systems) makes it improbable that our local system should be 
so nearly at rest in the aether. 



" Phil. Mag.," February, 19i8. 



CHAPTEB II 



THE RELATIONS OF SPACE, TIME AND FOKCE. 

9. An interesting aspect of the transformation of the 
variables x, y, z, t to x', y', z', t' has been brought out by 
MinkowskL We consider them as co-ordinates in a four- 
dimensional continuum, of space and time. Choose the units 
of space and time so that the velocity of light is unity, and set 

t=ir, where i=v/ L 
The equations of transformation (4*2) become 

(9-1) 



Let u=i tan 0, so that 6 is an imaginary angle. Then 
/?=cos 6, and (9*1) becomes 



r'sin 0, y=y', z~z', r=r / cos 0+s/sin . (9-2) 

Thus the transformation is simply a rotation of the axes of 
co-ordinates through an imaginary angle 6 in the plane of XT. 

We know that the orientation chosen for the space-axes, 
a?, /, 2, makes no difference in Newtonian mechanics. The 
principle of relativity extends this so as to include the axis r. 
The continuum formed of space and imaginary time is perfectly 
isotropic ; the resolution into space and time separately, which 
depends on the motion of the observer, corresponds to the 
arbitrary orientation in it of a set of rectangular axes. 

10. From this point of view the strange conspiracy of the 
forces of Nature to prevent the detection of our absolute 
motion disappears There is no conspiracy of concealment, 
because there is nothing to conceal. The continuum being 
isotropic, there is no orientation more fundamental than any 
other ; we cannot pick out any direction as the absolute time 
any more than we can pick out a direction in space as the 
absolute vertical, Up-and-down, right-and-left, backwards- 



THE RELATIONS OF SPACE, TIME AND FORCE. 15 

and-forwards, sooner-and-later,* equally express relations to 
some particular observer, and have no absolute significance. 
In Minkowski's famous words, 6S Henceforth Space and Time 
in themselves vanish to shadows, and only a kind of union of 
the two preserves an independent existence." 

The scientific basis of the idea that some fundamental 
division into space and time exists was the conception of the 
aether as a material fluid, filling uniformly and isotropically a 
particular space. It now seems clear that the aether cannot 
have those material properties which would enable it to serve 
as a frame of reference. Its functions seem to be limited to 
those summed up in the old description " the nominative of 
the verb ' to undulate.' " 

Unfortunately the simplicity of this conception of the four- 
dimensional continuum is only formal ; and natural pheno- 
mena make a discrimination between r and the other variables 
by relating themselves to an imaginary r, which we call the 
time. In natural variables, x, y, z, t, this view of the trans- 
formation as a rotation of axes becomes concealed. f 

11. In the four-dimensional continuum the interval ds 
between two point-events is given by 

. . . (11-1) 



which is unaffected by any rotation of the axes, and is therefore 
invariant for all observers. The minus sign given to ds* is an 
arbitrary convention, and the formula is simply the generalisa- 
tion of the ordinary equation 



The fact that ds is measured consistently by all observers 
who would obtain discordant results for 8x 9 dy, dz, dr separately, 
is so important in our subsequent work that we shall consider 
the nature of the clock-scale needed for its measurement. 

We have a scale AB divided into kilometres, say, and at 
each division is placed a clock also registering kilometres. 

* This applies to imaginary time. With real time, events which (as 
usually happens) are separated by a greater interval in time than in space 
preserve the same order for all observers. But an event on the sun which 
we should describe as occurring 2 minutes later than an event on the earth 
might be described by another observer as 2 minutes earlier. (Both ob- 
servers have corrected their observations for the light-time.) 

t For a logical study of the properties of the continuum of space and real 
time reference may be made to A. A. Robb, " A Theory of Time and Space " 
(Camb. Univ. Press). 

C 2r 



16 BELATIVITY THEORY OF GRAVITATION. 

(Tie velocity of light being unity, a kilometie is also a unit of 
time = a-obVoo" sec.) When, the clocks are correctly set and 
viewed from A, the sum of the readings of any clock and the 
division beside it is the same for all, since the scale-reading 
gives the correction lor the time taken by light in travelling 
to A. This is shown in Tig. 2, where the clock-readings are 
given as though they were being viewed from A. 

Now lay the scafe in line with the two events ; note the 
clock and scale-reading, 1? ^ 1? of the first event, and the 
corresponding readings ^ 2 > ^ of tte second event ; then from 
(11-1) 

a^l.-^M''.--^) 1 .... (11-2) 

If the scale had been set in motion in the direction AB 9 
<r a -~ "i would have been diminished,, owing to the divisions 
having advanced to meet the second event. But the clocks 
would have been adjusted differently, because A is now 




Fio. 2. 

advancing to meet the light coming from any clock, and the 
clock would appear too fast (by the above rule) if it were not 
set back. There are other second-order corrections arising 
from the contraction of the scale and change of rate of the 
clocks owing to motion ; but the net result is a perfect com- 
pensation, and ds z determined from (11-2) must be invariant, 
as already proved. 

It is clear that the whole (restricted) principle of relativity 
is summed up in this invariance of ds, and it is possible to 
deduce the equation of transformation (4 2) and our other 
previous results by taking this as postulate. 

When 8s refers to the interval between two events in the 
Itistory of a particular particle it has a special interpretation 
which deserves notice. If we choose axes moving with the 
particle, dx 9 dy> <5z==0, so that ds=*dt. Accordingly the variable 
s is called the " proper-time," i.e., the time measured by a 
clock attached to the particle, 

12. Up to the present we have discussed a particular type 
of transformation of co-ordinates, viz., that corresponding to 



THE RELATIONS OF SPACE; TIME AND FORCE. 17 

a uniform motion of translation. We now enter on the theory 
of more general changes of co-ordinates 

The co-ordinates x, y, z, t of a particle trace a curve in lour 
dimensions which is called the world-line of the particle. If 
we draw the world-lines of all the particles, light-waves and 
other entities, we obtain a complete history of the configurations 
of the Universe for all time. But such a history contains a 
great deal that is necessarily outside experience. All exact 
observations are records of coincidences of two entities in 
space and time, that is to say 3 records of intersections of world- 
lines. 

It is easy to see that this is the case in laboratory experi- 
ments or astronomical observations. Electrical measurements, 
determinations of temperature, weight, pressure, &c., rest 
finally on the coincidence of some indicator with a division 
on a scale. Many of our rough observations depend on co- 
incidences of light waves with elements of the retina, or the 
simultaneous impact of sound-waves on the ear. It is true 
that some of our external knowledge is not obviously of this 
character. We estimate the weight of a letter, balancing it 
in the hand ; this is based on a muscular sensation having no 
immediate relation to time and space, but we fit this crude 
knowledge into the exact scheme of physics by comparing it 
with more accurate measures based on coincidences. 

The observation that the world-lines of two particles intersect 
is a genuine addition to knowledge, since in general lines in 
space of three or four dimensions miss one another. We have 
to build up our conception of the location of objects in space 
and time from a large number of records of coincidences. It 
is clear that we have a great deal of liberty in drawing the 
world-lines, whilst satisfying all the intersections. Let us draw 
the world-lines in some admissible way, and imagine them 
embedded in a jelly. If the jelly is distorted in any way, the 
world-lines in their new courses will still agree with observation, 
because no intersection is created or destroyed.* 

Mathematically this can be expressed by saying that we may 
make any mathematical transformation of the co-ordinates. If 
we choose new co-ordinates x', y', z' } t', which are any four 
independent functions of x, y, z, t, a coincidence in x, y, z, t 
will also be a coincidence in. x', y', %', t', and vice versa. By 
locating objects in the space-time given by x' y y', z', t', we do 
not alter the course of events. The events themselves do not 
presuppose any particular system of co-ordinates, and the 



18 RELATIVITY THEOBY OF GRAVITATION. 

space-time scaffolding is something introduced arbitrarily by 
ourselves. 

It is almost a truism to say that we may adopt any system 
of co-ordinates we please. We are accustomed to introduce 
curvilinear co-ordinates or moving axes without apology, 
whenever they simplify the problem. But there is one point 
not so generally recognised. Ordinarily when we use curvi- 
linear co-ordinates we never allow ourselves to forget that 
they are curvilinear ; it is a mathematical device, not a new 
space, that we adopt. Perhaps the only case in which we 
really take the new co-ordinates seriously is in the trans- 
formation to rotating axes ; we then take account of the 
rotation by adding a fictitious centrifugal force to the equations, 
and thenceforth the rotation is quite put out of mind. From 
the standpoint of relativity, when we adopt new co-ordinates 
'> y'y z '> t', we shall adopt a corresponding new space, and think 
no more of the old space. For instance, a " straight line " in 
the new space will be given bv a linear relation between 

*, y', *, f. ^ 

The behaviour of natural objects will no doubt appear very 
odd when referred to a space other than that customarily used. 
So-called rigid bodies will change dimensions as they move ; 
but we are prepared for that by our study of the Michelson- 
Morley contraction. Paths of moving particles will for no 
apparent reason deviate from the " straight line," but, accept- 
ing the definition of a force as that which changes a body's 
state of rest or motion, this must be attributed to a field* oi 
force inherent in the new space (cf. the centrifugal force). 
Light-rays will alfeo be deflected, so that the field of force acts 
on light as well as on material particles ? this is not altogether 
a novel idea, because a little reflection shows that the centri- 
fugal force deflects light as well as matter although optical 
problems are not usually treated in that way. 

13. The laws of mechanics and electrodynamics are usually 
enunciated with respect to " unaccelerated rectangular axes," 
or, as they are often called, " Galilean axes." We cannot 
regard such axes as recognisable intuitively, and the only 
definition of them that can be given is that they are the axes 
with respect to which that particular form of the laws holds. 
It is part of the method of the present theory to restate the 
laws of Nature in a form not confined to Galilean co-ordinates, 
so that all systems of co-ordinates are regarded as on the same 
footing. 



THE KELATIONS OF SPACE, TIME AND FORCE. 19 

In unaccelerated rectangular co-ordinates the path of a 
particle is a straight line (apart from the influence of other 
matter, or the electromagnetic field)* When we transform to 
other co-ordinates the path is no longer straight, i.e., it is no 
longer given by a linear relation between the co-ordinates ; 
and the bending of the path is attributable to a field of force 
which comes into existence in the new space. This field of 
force has the property that the deflection produced is inde- 
pendent of the nature of the body acted on, being a purely 
geometrical deformation. Now the same property is shared 
by the force of gravitation the acceleration produced by a 
given gravitational field is independent of the nature or mass 
of the body acted on. This has led to the hypothesis 
that gravitation may be of essentially the same nature 
as the geometrical forces introduced by the choice of 
co-ordinates. 

This hypothesis, which v/as put forward by Einstein, is 
called the Principle of Equivalence. It asserts that a gravita- 
tional field of force is exactly equivalent to a field of force introduced 
by a transformation of the co-ordinates of reference, $o*that by no 
possible experiment can we distinguish between them. 

In Jules Verne's story, " Round the Moon/ 5 three men are 
shot up in a projectile into space. The author describes their 
strange experiences when gravity vanishes at the neutral point 
between the earth and moon. Pedantic criticism of so de- 
lightful a book is detestable ; yet perhaps we may point out 
that, for the inhabitants of the projectile, weight would vanish 
the moment they left the cannon's mouth. They and their 
projectile are falling freely all the time at the same rate, and 
they can feel no sensation of weight. They automatically 
adopt a new space, referred to the walls and fixtures of their 
projectile instead of to the earth. Their axes of reference are 
accelerated falling towards the earth ; and this transforma- 
tion of axes introduces a field of force which just neutralises 
the gravitational field. But, whilst they could jdetect no 
gravitational field by ordinary tests, it is not obviously im- 
possible for them to detect some effect by optical or electrical 
experiments. According to the principle of equivalence, how- 
ever, no effect of any kind could be detected inside the pro- 
jectile ; the gravitational field cannot be differentiated from a 
transformation of co-ordinates, and therefore the same^ trans- 
formation which neutralises mechanical efieots neutralises all 
other effects. 



20 RELATIVITY THEORY OF GBAVITATTON. 

It will be seen that this principle of* equivalence is a natural 
generalisation of the principle of relativity. An occupant of 
the projectile can only observe the relations of the bodies inside 
to himself and to each other. The supposed absolute accelera- 
tion of the projectile is just as irrelevant to the phenomena as 
a uniform translation is. The mathematical space-scaSoldmg 
of Galilean axes, from which we measure it, has no real 
significance. If the projectile were not allowed to fall, gravity 
would be detected or rather the force of constraint which 
prevents the fall would be detected. I think it is literally true 
to say that we never feel the force of the earth's attraction on 
our bodies ; what we do feel is the earth shoving against our 
feet. 

14. A limitation of the Principle of Equivalence must be 
noticed. It is clear that we cannot transform away a natural 
gravitational field altogether. If we could, we should un- 
consciously make the transformation and adopt the new co- 
ordinates just as the inhabitants of the projectile did. They 
were concerned with a practically infinitesimal region, and 
for an infinitesimal region the gravitational force and the 
force due to a transformation correspond ; but we cannot find 
any transformation which will remove the gravitational field 
throughout a finite region. It is like trying to paste a fiat 
sheet of paper on a sphere, the paper can be applied at any 
point, but as you go away from the point you soon come to a 
misfit. For this reason it will be desirable to define the exact 
scope of the principle of equivalence. Up to what point are 
the properties of a gravitational field and a transformation 
field identical ? And what properties does a gravitational 
field possess which cannot be imitated by a transformation ? 
The impossibility of transforming away a gravitational field is, 
of course, an experimental property ; so that, in spite of the 
principle of equivalence, there is at least one means of making 
an experimental distinction- 

Space-time in which there is no gravitational field which 
cannot be transformed away is called homaloidal. In homa- 
loidal space-time then, we can choose axes so that there is no 
field of force anywhere. Remembering that we have no means 
of defining axes except from the form of the laws of Nature 
referred to them, we should naturally take these axes as 
fundamental and name them ** rectangular and unaccelerated." 
The dynamics of homaloidal space would not recognise the 
existence of gravitation. Our space is not like that, though 



THE RELATIONS OF SPACE, TIME AND FORCE. 21 

we believe that at great distances from all gravitating matter 
it tends towards this condition as a limit. The necessary 
limitation of the principle ot equivalence turns on the number 
of consecutive points for which gravitational space-time agrees 
with homaloidal space-time ; in other words, the equivalence 
will hold only up to a certain order of differential coefficients. 
Properties involving differential coefficients up to this order 
will be the same in the gravitational field as in a homaloidal 
field ; whilst properties of the transformed field involving 
differential coefficients of higher order will not necessarily hold 
in the gravitational field 

The determination of the order of the differential coefficients 
fo: which agreement is possible must be deferred to 27. 
Meanwhile it may be noted that we can always choose axes 
for which the field at a given point vanishes viz., take rect- 
angular axes moving with the acceleration at that point. In 
that case we are said to use " natural measure." 

15. At a point of space where there is no field of force the 
observer's clock-scale, if unconstrained, will be either at rest 
or in uniform motion. We have seen that the measured 
interval, ds, between two events is independent of uniform 
motion, and hence a unique value of ds is determined by the 
measures. 

Using rectangular co-ordinates, the relation between an 
infinitesimal measured interval ds and the inferred co-ordinates 
of the event is (11-1). 



. . . (15-1) 

Introduce new co-ordinates x v x& x& x& which are any 
functions of x, y, z> t given by 



Tim ^= 1+ -l-^ 8 + 4) 40. . (15-2) 



Substituting (15-2) on the right-hand side of (15-1), we 
obtain a general quadratic function of the infinitesimals, which 
may be written, 

dx 2 dx^2g^dx^dx^ . (15-3) 

where the y's are functions of the co-ordinates, depending on 
the transformation, 



22 BELAT1VITY THEORY OF GRAVITATION. 

As n illustration we may take the transformation to rotating 
axes 

XX-L cos (DX -x 2 sin 
yx^ sin a># 4 -j-x. 2 cos 

Whence 

<#:r cos a>x^dx l sin oyx^.dx^ &>(#i sin a>x^-\-x^ cos 
$y^= sm cox^dx^-^ cos cosc4,d#2~j- ft)(Xj cos o># 4 # 2 sin 



Substituting in (15-1) 

ds~ == <fcr* foa rfa/3 -f- (1 o> 2 (: 

.& CO*Cj[GJ2?2^*^4 ** (lc)~D/ 

By comparing this with (15 3) we obtain the values of the gr's 
for this system of co-ordinates. 

16. These values of the gr's express the metrical properties of 
the space that is being used. But the observer has no im- 
mediate perception of them as properties of space. He does 
not reaHse that there is anything geometrically unnatural 
about axes rotating with the earth, but he perceives a field of 
centrifugal force. Experiments, such as Foucault's pendulum 
and the gyro-compass, designed to exhibit the absolute 
rotation ot the earth, are more naturally interpreted as de- 
tecting this field of force. 

Thus the coefficients g ll9 &c., can be taken as specifying a 
field of force. That they are sufficient to define it completely 
may be seen from the following consideration. The world-line 
of a particle under no forces is a straight line in the system 
a?, ?/, 2, , and its equation may be written in the form 



r is stationary ; (16*1) 

but in this form the equation is independent of the choice 
of co-ordinates, and applies to all systems. If we choose 
new co-ordinates, the world-line given by (16*1) becomes 
curved and the curvature is attributed to the field of force 
introduced ; but clearly the curvature of the path can only 
depend on the expression for ds in the new co-ordinates, t.e., 
on the gr's. Thus the force is completely defined by the y's. 
It will be noticed that in (15-5) 



(16-2) 

where Q=|co^(^^+^2) r:: =the potential of the centrifugal force* 



THE EELATIONS OF SPACE, TIME AND FORCE. 23 

Thus # 4i can be regarded as a potential ; and by analogy 
all the coefficients are regarded as components of a generalised 
potential of the field of force. 

According to the principle of equivalence it must also be 
possible to specify a gravitational field by a set of values of 
the 0's. It will be our object to find the differential equations 
satisfied by the g's representing a gravitational field. These 
differential equations for the generalised potential will express 
the law of gravitation, just as the Newtonian, law is expressed 
by y2 9 =0. 

The double aspect of these coefficients, gr tl , &c., should be 
noted. (1) They express the metrical properties of the co- 
ordinates. This'is the official standpoint of the principle of 
relativity, which scarcely recognises the term " force." 
(2) They express the potentials of a field of force. 
This is * the unofficial interpretation which we use when 
we want to translate our results in terms of more familiar 
conceptions. 

Although we deny absolute space, in the sense that we regard 
all space-time frameworks in which we can locate natural 
phenomena as on the same footing, yet we admit that space 
the whole group of possible spaces may have some absolute 
properties. It may, for instance, be homaloidal or non- 
homaloidal. Whatever the co-ordinates, space near attracting 
matter is non-homaloidal, space at an infinite distance from 
matter is homaloidal. You cannot use the same co-ordinates 
for describing both kinds of space, any more than you can use 
rectangular co-ordinates on the surface of a sphere ; that is, 
in fact, the geometrical interpretation of the difference. 
Homaloidal space-time may be regarded as a four-dimensional 
plane drawn in a continuum of five dimensions ; whereas 
non-homaloidal space-time must be regarded as a curved 
surface in five dimensions.* These considerations apply, of 
course, to measured space ; we can always throw the^ blame 
on our measuring rods, and apply theoretical corrections to 

* We shall see (44) that m a region, not containing matter, but traversed 
by a gravitational field due to matter, the Gaussian or total curvature is 
zero ; but such a space-time does not correspond to a plane m five dimen- 
sions, or to any surface whioh can be developed into a plane. The space- 
time m a gravitational field has an essential curvature m the ordinary 
sense, although it happens that the particular invariant technically called 
" the'curvature " vanishes. In three-dimensional space a surface with zero 
Gaussian curvature can always be developed into a plane ; but this is not 
true for space of higher dimensions, so that the three-dimensional analogy 
is liable to lead to misimderstanding. 



24 RELATIVITY THEORY OF GRAVITATION. 

OUT measures so as to make them agree with any kind of apace 
we please. 

It is not necessary, and indeed it is not possible, to draw a 
sharp distinction between the portions of the ^'s arising from 
the choice of co-ordinates and the portions arising from the 
gravitation of matter. We have seen that, when there is no 
field of force, ds* has the form (15-1), so that the #'s have the 



looo 

0-100 
00-10 
0001 



. . . . (16-3) 



These values then express that there is no field of force, 
and in the absence of a gravitational field produced by matter 
it is possible to take our co-ordinates (Galilean co-ordinates) 
so that the values (16-3) hold everywhere. We naturally 
regard such co-ordinates as fundamental ; and, if we choose 
any other co-ordinates, the deviations of the /s from this 
peculiarly simple set of values are regarded as due to the 
distortion of the space-time chosen But by 14 ; when 
gravitating matter is in the neighbourhood, there is no possi- 
bility of choosing co-ordinates, so that the values (16-3) hold 
everywhere, and there is no criterion for selecting any one of 
the possible systems of co-ordinates as more fundamental than 
the others. * 

Accordingly we shall henceforth apply the term " gravita- 
tional field " to the whole field of force given by the #'s, what- 
ever its origin. In the particular case when no part of it is 
due to the gravitation of matter, we shall say there is no 
permanent gravitational field. 

Just as Galilean co-ordinates are defined by the values (16-3) 
of the y's, so any other co-ordinates must be defined analytically 
by specifying the j's as functions of x v x z , x 3y cc 4 , or what 
comes to the same thing by giving the expression for cfe 2 . 
For example, if in Ijwo dimensions dfP=dx I *+x I *dx, the 
co-ordmates axe recognised as plane polar co-ordinates with 

* Thus if we say " take rectangular axes with the sun as origin " the 
statement is ambiguous Unaccepted rectangular axes imply that & ia 
of the form (15-1) no other means of defining them having yet been given. 
Owing to the sun s gravitation there is no system of co-ordinates for which 
this is true, and several different systems present rival claims to be regarded 
as the best approximation possible. The di fficulty does not arise if we only 
have to consider an infinitesimal region of space ; in that case the co-ordinates 
(giving natural measure ") are defined without ambiguity. 



THE RELATIONS OF SPACE, TIME AND FORCE. 25 



a^r, x 2 =6 ; if ds^^dx^+cos^dx^, the co-ordinates are 
latitude (a^) and longitude (a? 2 ) on a sphere. We might take 
for the 10 gf's perfectly arbitrary functions of x l3 x 2 , o? 3 , x A9 
and so obtain a ten-fold infinity of mathematically conceivable 
systems of co-ordinates. But this would include many systems 
of co-ordinates which describe kinds of space-time not occurring 
in Nature. In any particular problem our choice is restricted 
to a four-fold infinity, viz., if x^ o? 2 , x& # 4 is a possible system, 
then four arbitrary functions of x v x 2 , x 39 x will form a possible 
system. In some other problem there will be an entirely 
different group of possible systems ; the space-times in the 
two problems have thus certain absolute properties which are 
irreconcilable, and we interpret this physically by saying that 
the permanent gravitational field is different in the two cases. 
Further, taking all possible distributions of permanent gravi- 
tational field which can occur in space (in the neighbourhood 
of, but not containing, matter), we do not exhaust the con- 
ceivable variety of functions expressing the gr's. There is a 
general limitation on the </'s imposed, not by mathematics, 
but by Nature which is expressed by the differential equa- 
tions of the law of gravitation which we are about to seek* 
The law of gravitation, in fact, expresses certain absolute 
properties common to all the measured space-times that can 
under any conditions occur in Nature. 

The law of gravitation, or general relation connecting the 0's, 
must hold for all observed values of the #'s. iSince the f/'s 
define the system of co-ordinates used, this means that the 
relation must hold for all possible systems of co-ordinates. 
If new co-ordinates are chosen, we find new values of the gr's 
as in (15*5) ; and the differential equations between the new 
0's and new co-ordinates must be the same as between the old 
<7*s and old co-ordinates. In mathematical language the equa- 
tions must be covariant. 

There is a resemblance between this statement and the 
statement of 12 which is somewhat deceptive. We there 
found that observable events have no reference to any parti- 
cular system of co-ordinates, and therefore all laws of nature 
can be expressed in a form independent of the co-ordinates. 
But this alone does not allow us to deduce tlie covariance of 
the equations satisfied by the gravitation-potentials. Without 
the principle of equivalence we could no doubt define the field 
by certain potentials cp^ <p 2> 93, . . . . which satisfy differential 
equations independent of the choice of co-ordinates. But that 



26 RELATIVITY THEORY OF GRAVITATION. 

conveys no information of value, unless we are told how to 
find <PJ, <p a ', .... in the co-ordinates a, a?,, a?;, a? 4 ' from the 
values <p l5 9 2 , .... in the co-ordinates x ly # 2 , # 3 , # 4 . The 
statement in 12 tells us nothing about that. It is the prin- 
ciple of equivalence which, by identifying the potentials with 
the #'s for which the method of transformation is known, 
supplies the missing link 

17. The Newtonian law of gravitation, V 2 #44=^> does not 
fulfil the condition of covanance nor does any modification of 
it, which immediately suggests itself. We have, therefore, to 
seek a new law guided by the condition that it must be ex- 
pressed by a covariant set of equations between the y's. It 
will be found in Chapter IV. that the choice is so restricted 
as to leave little doubt as to what the new law must be. 

If we write the required equations in the form 

^=0, y 2 =0, T 3 =0, Ac., 

the left-hand sides, T 13 T 2 , T 3 , may be regarded as components 
of a kind of generalised vector, only the number of components 
is not, as in a vector, restricted to 4. 

The covariance of the equations means that, if all the com- 
ponents vanish in one system, of co-ordinates, they must vanish 
in all systems. To secure this, T 19 T& . . . . must obey a 
linear law of transformation ; thus 



where the coefficients are functions of the co-ordinates de- 
pending on the transformation. Generalised vectors of this 
kind are called tensors ; and it will be necessary for us to 
study their properties in the next chapter, in order to select 
the one which can represent the new law of gravitation. 

We see that if an equation is known to be a tensor-equation, 
it is sufficient to prove it for one particular system of co- 
ordinates ; it will then automatically hold in any other system 
obtainable by a mathematical transformation, 

The more general purpose of the tensor theory is this : 
If we are given a set of equations expressing some physical 
law in the usual co-ordinates, we may be able to recognise 
these as the degenerate form for Galilean co-ordinates of some 
tensor equation. Expressed in teasor form, these equations 
will then hold for all systems of co-ordinates that can be 
derived by a mathematical transformation. Subject to the 



16 BELATIVITY THEORY OF GRAVITATION. 

(Tie velocity of light being unity, a kilometie is also a unit of 
time = a-obVoo" sec.) When, the clocks are correctly set and 
viewed from A, the sum of the readings of any clock and the 
division beside it is the same for all, since the scale-reading 
gives the correction lor the time taken by light in travelling 
to A. This is shown in Tig. 2, where the clock-readings are 
given as though they were being viewed from A. 

Now lay the scafe in line with the two events ; note the 
clock and scale-reading, 1? ^ 1? of the first event, and the 
corresponding readings ^ 2 > ^ of tte second event ; then from 
(11-1) 

a^l.-^M''.--^) 1 .... (11-2) 

If the scale had been set in motion in the direction AB 9 
<r a -~ "i would have been diminished,, owing to the divisions 
having advanced to meet the second event. But the clocks 
would have been adjusted differently, because A is now 




Fio. 2. 

advancing to meet the light coming from any clock, and the 
clock would appear too fast (by the above rule) if it were not 
set back. There are other second-order corrections arising 
from the contraction of the scale and change of rate of the 
clocks owing to motion ; but the net result is a perfect com- 
pensation, and ds z determined from (11-2) must be invariant, 
as already proved. 

It is clear that the whole (restricted) principle of relativity 
is summed up in this invariance of ds, and it is possible to 
deduce the equation of transformation (4 2) and our other 
previous results by taking this as postulate. 

When 8s refers to the interval between two events in the 
Itistory of a particular particle it has a special interpretation 
which deserves notice. If we choose axes moving with the 
particle, dx 9 dy> <5z==0, so that ds=*dt. Accordingly the variable 
s is called the " proper-time," i.e., the time measured by a 
clock attached to the particle, 

12. Up to the present we have discussed a particular type 
of transformation of co-ordinates, viz., that corresponding to 



28 BBLATIVITY THE OB Y OB 1 OBAVITATION. 

But there is more than one way of correcting tte measures 
to fit Euclidean space, so that we are not really justified in 
making precise statements as to the behaviour of our clocks 
and measuring rods. It is better not to discuss their defects, 
but to accept the measures and examine the properties of the 
corresponding non-Euclidean space and time. 

If we draw a circle with a heavy particle near the centre, 
the ratio of the measured circumference to the measured 
diameter will be a little less than rr > owing to the factor y~l 
affecting radial measures. It is thus like a circle-drawn on a 
sphere, for which the circumference is less than ^ times the 
diameter if we measure along the surface of the sphere. We 
may imagine space pervaded by a gravitational field to have 
a curvature in some purely mathematical fifth dimension. 




If we draw the elliptic orbit of a planet, slit it along a radius 
and try to fold it round our curved space there will evidently 
be some overlap. For example, take a cone with the sun as 
apex as roughly representing the curved space. Starting with 
the radius vector SP, the Euclidean space will fold completely 
round the cone and overlap to the extent PSP\ Thus the 
corresponding radius advances through an angle PSP' each 
revolution (Fig. 3). This shows one reason for the advance of 
perihelion of a planet, which is one of the most important 
effects predicted by the new theory ; but it is not the whole 
explanation. 

The reader may not unnaturally suspect that there is an 
admixture of metaphysics in a theory which thus reduces the 
gravitational field to a modification of the metrical properties 
of space and time. This suspicion, however, is a complete 
misapprehension, due to the confusion of space, as we have 
defined it, with some transcendental and philosophical space. 



THE RELATIONS OF SPACE, TIME AND FORCE. 29 

There is nothing metaphysical in the statement that imder 
certain circumstances the measured circumference of a circle 
is less than n times the measured diameter ; it is purely a 
matter for experiment. We have simply been studying the 
way in which physical measures of length and time fit together 
just as Maxwell's equations describe how electrical and 
magnetic forces fit together. The trouble is that we have 
inherited a preconceived idea of the way in which measures, 
if " true/ 9 ought to fit. But the relativity standpoint is that 
we do not know, and do not care, whether the measures under 
discussion are " true " or not ; and we certainly ought not to 
be accused of metaphysical speculation, since we confine our- 
selves to the geometry of measures which are strictly practical, 
if not strictly practicable. It is desirable to insist that we do 
not attribute any causative properties to these distortions of 
measured space and time. To hold that a property of our 
measuring-rods is the cause of gravitation would be as absurd 
as to hold that the fall of the barometer is the cause of the 
storm. 



CHAPTER III. 



THE THEORY OF TENSORS. 

19. We consider transformations from one system of co- 
ordinates # 1? o: 2 , # 3 , #4 to another system a^, 2,', x, x** 

(a) Notation. 

The formula (15*3) for d$ 2 may be written 

4 4 

x,, (g v =g^) . . . (19-11) 



In the following work we shall omit the signs of summation, 
adopting the convention that, whenever a literal suffix appears 
twice in a term, the term is to be summed for values of the 
suffix 1, 2, 3, 4. If a suffix appears once only, no summation 
is indicated. Thus we shall write (19*11) 

ds^^g^dx^dxy ...... (19*12) 

In rare cases it may be necessary to write a term containing 
a suffix twice which is not to be summed ; these cases will 
always be specially indicated. In general, however, this con- 
vention anticipates our desires, and actually gives a kind of 
momentum in the right direction to the analysis. 

As a rule of manipulation it may be noticed that any suffix 
appearing twice is a dummy, and can be changed freely to any 
other suffix not occurring in the same term. 

(6) Covariant and Contravariant Vectors. 
The vector (dx ly dx& dx%, dx) is transformed according to 
the equations 



or, with our convention as to notation 



THE THEOBY OF TENSORS. 31 

Any vector transformed according to this law is called a 
contravanant vector : its character is denoted by the notation 
JM (ji*=l, 2, 3, 4). The law may be written 

(19-21) 

where, as already explained, summation is indicated by the 
double appearance of the dummy <r. 
If 9 is a scalar (i.e., invariant) lunction of position the vector 

-- ~~- 2- -2- is transformed according to the law 

^9. i 3#2 d<? . d^a ^9 . 9as 4 _9q 



A vector transformed according to this law is called a 
covanant vector, denoted by A^ The law may be written 



4;= 4, ...... (19-22) 

0#> 

A covariant vector is not necessarily the gradient of a scalar. 

The customary geometrical conception of a vector does not 
reveal the distinction between the two classes of contravariant 
and covariant vectors. We usually represent any directed 
quantity by a straight line, which should strictly correspond 
only to" a contravariant vector. The other class of directed 
quantities is more properly represented by the reciprocal of a 
straight line ; but in elementary applications, when we are 
thinking in terms of rectangular co-ordinates, there is no need 
to make this distinction. Consider, however, a fluid with a 
velocity potential. With rectangular co-ordinates the velocity 
is equal to the gradient of the velocity potential. Both these 
are directed quantities, i.e., vectors, and the vector relation 
extends to their rectangular components ; thus 
dx _ 89 dy _ 9<p dz _ 9<p 
di^fo 9 dt~"<h/ dt~~dz~ 

But if we use oblique axes or curvilinear co-ordinates, the 
relation no longer holds. E.g., it is not true that in polar co- 
ordinates dQ/dt=dq>/dd ; the actual relation is rd6/dt=d(pfrd6. 
This is because the two vectors are of opposite natures, the 
first being contravariant and the second covariant. K they 
tad been of the same nature the relation must have held for 
all systems, of co-ordinates. Clearly, since in our work we 
consider all systems of co-ordinates as on the same footing, 
*e have to distinguish carefully between the two types. 

D2 



32 BELATIVITY THEORY OF GRAVITATION, 

realise at once that the equation d^/afc==d<p/9a^, being an 
equation between vectors of opposite kinds, is impossible as a 
general equation for all systems of co-ordinates, i.e., it is not 
a covariant equation. 

(c) Tensors of Higher Rank. 

We can denote by A^ a quantity having 16 components, 
obtained by giving different numerical values to ju and ?. 
Similarly, A^ has 64 components. By a generalisation of 
(19-21) and (19-22) we classify quantities of this kind according 
to their transformation laws, viz., 

Covariant tensors J^---^^^ . . . . (19-31) 

Contravariant tensors A^ v =^-~-^A ar . . . , (19-32) 

Mixed tensors ^'^^tix^* ( 19 * 33 ) 

and similarly for tensors of the third and higher rank. These 
equations of transformation are linear, so that the conditions 
of 17 are satisfied. Also it is not difficult to see that there 
can be no other linear types of transformation-laws having the 
necessary transitive property. For example, consider a vector 
A r , Introducing a third set of co-ordinates x, we have 




^ , 

But 

showing that the result is the same whether the transformation 
is performed in two steps or directly. Other suggested types 
of transformation law have not this necessary property. Thus 
all possible types of tensors are included. 

Evidently the sum of two tensors of the same character is a 
tensor. 

The product of two tensors is a tensor, and its character is 
the sum of the characters of the component tensors. For 
example, consider the product A^B^w have by (19-31) and 



He... 



THE THEORY OF TENSORS. 33 

showing that the law of transi oimation is that of a tensor of 
the fourth rank having the character denoted by C^. 

The product of two vectors is a tensor of the second rank, 
but a tensor of the second rank is not necessarily the product 
of two vectors. 

A familiar example of a tensor of the second rank is afforded 
by the stresses in a solid or viscous fluid. The component of 
stress denoted by p xy represents the traction In the ^-direction 
exerted across an interface perpendicular to the ^-direction. 
Bach component involves a specification of two directions. 

(d) Inner MuU^phcat^on. 

If we multiply A^ by B*, the repetition of the suffix involves 
summation of the resulting products. The result is called the 
inner product in contrast to the ordinary or outer product A^B V . 
The notation at once shows whether the product is inner or 
outer in any formula. 

From a mixed tensor such as A r ^ vcr we can form a " contracted " 
tensor A^ v<r , which is of the second rank with suffixes ft and v 
(since v is now a dummy suffix). To show that it is a tensor 
we have as in (19*34) 



But ^^=5-^=0 or 1, according as y / d or y=d. 
ox* o%6 ox& 

Hence g^U^+0+0+^ r 

Substituting in (19*41) we see that A^ vff follows the law of 
transformation (19-31) and is therefore a co variant tensor. 

An expression such as A^ is not a tensor, and no interest 
attaches to it. 

By a similar argument we see that A% 3 A* v are invariant, 
and consequently A^B^ is an invariant. An invariant, or 
scalar, corresponds to a tensor of zero rank. 

(e) Criterion for the Tensor Character. 

To prove that a given quantity is a tensor, we either find 
directly its equations of transformation, or we express it as 
the sum or product of other tensors, or, under certain re- 
strictions, as the quotient of two tensors according to the 
following theorem : A quantity, which on inner multiplication 
by any covariant (alternatively, by any contravariant) vector 
always gives a tensor, is itself a tensor. 



34 RELATIVITY THEORY OF GRAVITATION. 

To prove this, suppose that A^ V B V is a covariant vector for 
any choice of the contravariant vector B v . Then by (19-22) 



But by (19*21) applied to the inverse transformation from 
accented to unaccented letters, 

*-',*' 



Since B'" is arbitrary, the quantity in the bracket must 
vanish, showing that A^ v is a covariant tensor (19-31). The 
proof can evidently be extended to tensors of any character. 

20, (a) The Fundamental Tensor. 

Since g^dx^dx^ds*, which is an invariant or tensor of zero 
order, and dx v is an arbitrary eontravariant vector, it follows 
from the last theorem that g^dx^ is a covariant tensor of 
the first rank. Repeating the argument, since dx^ is an 
arbitrary eontravariant vector, g^ v must be a covariant tensor 
of the second rank. 

The determinant formed with the elements g^ v is called the 
fundamental determinant and is denoted by g. 

We define g^ to be the minor of g^ y divided by g. 

From this definition g^g^ reproduces the fundamental 
determinant divided by itseli, when o-=v, and gives a deter- 
minant with two rows identical, when <r*'v. We write 

gZ^g^g^l when o- - v \ 

=0 when <H=* * * 

Hence if A v is an arbitrary eontravariant vector 

$^=^+0+0+0=^ .... (20-15) 

This shows by the theorem of 19 (e) that g is a tensor, 
and it evidently is a mixed tensor as the notation has antici- 
pated.* 

* In applying the theorem of 19(e), the appropriate notation for the 
tensor* (expressing its oovariant or eontravariant character) is found by 
inspection. An equation such as (20*15) must have the suffixes on both 
uidfis In corresponding positions ; the upper and lower <r on the left cancel 
one anoiheil (7/. equations (20*21), (20-22), (20-23). It must be noted, 
however, tha& in an expression such as g^wdx^, dx^ is contra variant, so thai 
the second n is really an tipper suffix. 



THE THEORY OF TENSORS. 35 

(Similarly, since g^A a is a covariant vector, arbitrary on 
account of the free choice of A* 9 and fg*A*A^ g* must 
be a contravariant tensor. 

We Lave thus the three fundamental tensors 

SU yl, and $T, 

of covariant, mixed and contravariant characters. 

It will be seen from (20-15) that gl acts as a substitution 
operator substituting v for <r in the operand. 

(6) Associated Tensors. 

With any covariant tensor A MP we can associate 

a mixed tensor Al ^g v *A^ (20-21) 

a cent ravarianfc tensor A^q^g^A^g^Al- . . (20-22) 
a scalar A=g"A^AZ .... (20-23) 

(c) The Jacobian.- 

Denoting the determinant formed with elements a^ by 
la^l, the Jaeobian of the transformation is 



since in our notation the ordinary rule for multiplying deter- 
minants is |X fi \ X | B Ay 1 = \ A^B y I (left side not summed). 

Hence (f^Jstl* 

It d'c is an element of four-dimensional volume, we have 



_ 
sottat V-gr.it^V-j'.^ .... (20-3) 

We shall always assume that the Jaeobian is finite, i.e., 
that the transformation has no singularity in the region con- 
rictered. The determinant g is always negative for real trans- 



36 RELATIVITY THEORY OF GRAVITATION, 

21. Auxiliary Formula for the Second Derivatives. 
We introduce certain quantities known as Christoffers 
3-index symbols, viz., 

, 91 in 

' (2M1) 



We have {/*v, Jl} -^ [JM^, a] .... (21-13) 

and the reciprocal relation follows by (20-1) 

L*v, i]=ftu {/^v, a} .... (21-14) 

Since ^ is a co variant tensor 

, 9a?g fa? 

$w o^.' o/?^' 

Heace S ^ 8x " 



In the second term in the bracket we have interchanged 
a and ft, which is legitimate since they are dummies ; in the 
last term we have used 

_3 __ dx y 8 

Similarly, &4~a**V 



where in the last term we have made some interchanges of 
the dummy suffixes a, /?, y. 
Adding these two equations and subtracting (21-15)' we have 



Multiply through by ff^ p ~ /3 we have 



(19-82) 



using (20-1) and (2143). 



THE THEORY OF TENSORS. {Tt 

This somewhat complicated formula for d*x,/fa'd3 in 
terms of the first derivatives is needed for the developments 
in the next paragraph. 

22. Covariant Differentiation. 

If we differentiate a scalar quantity we obtain a tensor (a 
covariant vector) ; but if we differentiate a tensor of the first 
or higher rank the result is not a tensor. We can, however, 
obtain a tensor which plays the part of a derivative by a more 
general process. The process is particularly useful in gene- 
ralising results which have been obtained in Gfalilean co- 
ordinates, since the simple derivative is the degenerate form 
for Galilean co-ordinates of the covariant derivative here 
considered. 

If Ap is a covariant vector, then by (19-22) 



Whence, differentiating, 



... _ , .._._ _ ,_ 7/1 

nX c*Xt nffi nX flff* nT 

Substitute for d*x<,/dxdx; by (21-2) ; we have 



-$ fa> {<*$><*} A* (22-1) 

But A ff *r-;=Ap by (19-22) ; and in the last term the dummies 
/* 

a, & <r may be replaced by or, r, p. Hence if we write 

A 3A 
A^^^-f {pv 9 p}A p .... (22-2) 

(jjjy 

W6 have 4^=^7 -4^.,., 

showing that A^ v is a tensor. This is called the covariant d& 
rivative of A^ 

If A xy B^ are covariant vectors, A Xv> B^ v their covariant de- 
rivatives, then A 



is the sum of two tensors, and is therefore a tensor. Sub- 
stituting from (22-2) this tensor becomes 

9 . .(22-3) 



38 RELATIVITY THEORY OF GRAVITATION. 

which is called tlie derivative of the tensor A X B^ It is not 
difficult to show that any tensor of the second rank can be 
expressed as the sum. of products of four pairs of vectors, and 
hence (22*3) can be generalised, giving for the covariant 
derivative of A^ 

{^e}A, e . . (224) 




In a somewhat similar manner formulae for the covariant 
derivatives of contravariant and mixed tensors can be ob- B 
tained, viz., 

(225) 

. . (22-6) 

- (227) 

The unsymmetrical behaviour of covariant and contra- 
variant indices in these formula should be noticed. In all 
cases differentiation adds one unit of covariant character. 

When the # 5 s have Galilean values (or, more generally, are 
constants) the Christoff el symbols vanish, and these derivatives 
reduce in all cases to the ordinary differential coefficients. 

23. The R^emann-Christoffel Tensor. 

Let us form the second covariant derivative of the vector A^ 
that is to say in formula (22-4) we give the tensor A^ the 
value (22-2). 



1 

+ {v<r, 6} {/*, pj 

The first five terms are unaltered by interchanging v and <r, 
i.e, 9 by changing the order of differentiation. (We can write 
e for p in the second term.) Hence 

A _ A _ 

"-,u,vir z - l /Mn 



, 4 {", Pi ~ (I*, } {" P} +- {^, ?} -- (I*, p} 



THE THEORY OF TENSORS. 39 

The left side is a tensor, and A p is an arbitrary covariant 
yector ; therefore, by 19 (e) the quantity in the bracket is a 
tensor. This is called the Eiemann-ChristofM tensor, and is 
denoted by 



24. Conditions for Vanishing of the Riemann-Christoffel 

Tensor. 

Prom the foregoing definition the primary meaning of the 
vanishing of this tensor is that the order of differentiation is 
indifferent (as in the ordinary differentiation). But the tensor 
has an even more important property. It will be seen on 
inspection that it vanishes when the y's have their constant 
Galilean values.* But, since it is a tensor, it must also vanish 
in any other system of co-ordinates derivable by a mathe- 
matical transformation. Thus the equation 

5^=0, ...... (24-1) 

is a necessary condition that with suitable choice of co-ordinates 
d$ 2 can be reduced to the form 



. . . (24-2) 

In other words it is a necessary condition for the absence of 
a permanent gravitational field. 

It can be shown that the condition is also sufficient. 
Equation (24-1) contains 96 apparently different equations, 
since, owing to the antisymmetry in or and v, there are only 6 
combinations of <r and v to be combined with 16 combinations 
of p and p. But these are not all independent, and the 
number can be reduced to 20, which can be shown to be the 
number of conditions required for the transformation to the 
form (24-2) to be possible. 

The reduction is effected by writing 



so that BL,=g(tAkn>) by (20-1) 

Equation (24-1) is thus equivalent to 

(/iT(Tv)=0, 

and vice versa. 

* The Christoffel symbols vanish when the #'s are constants. 



40 RELATIVITY THEORY OF GRAVITATION. 

On working out the value of (pr<rv) it is seen by inspection 
that the following additional relations exist : 



which reduce the number of independent conditions to 20. 

25. To sum up what has been accomplished in this chapter, 
we have discussed the theory of tensors expressions which 
have the property that a linear relation between tensors of 
the same character will hold in all systems of co-ordinates if it 
holds in one system. We have shown that the tensor-property 
can be established either by determining the law of transforma- 
tion, or exhibiting the quantity as a sum or product of other 
tensors, or, under certain restrictions, as the quotient of tensors. 
We have found formulae for tensors which play the part of 
derivatives. Finally, we have found the necessary and 
sufficient relation between the g^ v> which must be satisfied in 
all systems of co-ordinates, when there is no permanent 
gravitational field. $ 

This last result is an important step towards obtaining the 
law of gravitation. Any set of values of the #'s which satisfy 
(24-1) will correspond to a possible set of co-ordinates which 
can be used for describing space not containing a permanent 
gravitational field. Hence if (24-1) is satisfied the #'s are such 
as can occur in Nature, and are accordingly not inconsistent 
with the law of gravitation. The required equations of the 
law of gravitation must, therefore, include the vanishing of 
the Riemann- Christ off el tensor as a special case. 



CHAPTER IV. 



EINSTEIN'S LAW OP GRAVITATION. 

26. We liave seen in 16 that the law of gravitation must 
be expressed as a set of differential equations satisfied by the 
^'s. We have further found the equations (24 1) which are 
satisfied in the absence of (i.e., at an infinite distance from) 
attracting matter. Clearly the general equations between the 
g's must be covariant equations automatically satisfied when 
(24-1) is satisfied ; but they must be less stringent, so as to 
admit of permanent gravitational fields, which, we know, do 
not satisfy (24-1). 

The simplest set of equations that suggests itself is 

GU=5/, p ==0 ...... (26-1) 

&,,, being the contracted Riemaiin-ChristofEel tensor, formed 
by setting 0*= p and summing. It is evidently satisfied when 
all components of the Riemann-Christoffel tensor vanish ; and 
it is a less stringent condition. 

The equations G> are taken by Einstein for the Law 
of Gravitation. Written in full they are, by (23) 



(26-2) 
The last two terms can be simplified. We have ' 



the other terms cancelling on summation. 

Hence, since g pe g is the minor of the element g p9 in the 
determinant g, 



4:2 RELATIVITY THEORY OF GRAVITATION. 

Equation (26-2) thus becomes 




=0 . (26-3) 



The equation is symmetrical in ^ and V, and therefore 
represents 10 different equations. Actually there exist four 
identical relations between these, so that the number of in- 
dependent equations is reduced to six (see 39). 

The selection of this law of gravitation is not so arbitrary 
as it might appear. There is no other set of equations corre- 
sponding to a tensor of the second rank containing only first 
and second derivatives of the g^ and linear in the second 
derivatives. Moreover, there is no other way of building up a 
tensor of lower rank out of the components of -B v<r .* 

Having regard to the summations involved in (26*3) it will 
be seen that the application of the new law of gravitation 
must involve a considerable amount of calculation. There are 
first to be calculated 40 different Christoffel symbols, each of 
which is the sum of 12 terms. Then each of the 10 equations 
contains 25 terms chiefly products or derivatives of the 
Christoffel symbols. Finally the partial differential equations 
have to be solved. It will probably be admitted that it is 
worth while to find out whether this suggested law of gravita- 
tion will agree with observation before resorting to something 
more complicated. 

27. We are now in a position to define the Principle of 
Equivalence more precisely. The difference between a per- 
manent gravitational field and an artificial one arising from a 
transformation of Galilean co-ordinates is that in the latter 
case (24-1) is satisfied, whereas in the former the less stringent 
condition (26-1) is satisfied. These equations determine the 
second differential coefficients of the g^ so that we can make 
the natural and artificial fields correspond as far as first 
differential coefficients, but not in the second differential co- 
efficients. We shall therefore state the Principle of Equiva- 
lence as follows : 

* The tensor S^ V<T vanishes identically. Other suggestions such as 
gwjfpvff merely give a set of equations equivalent to (26-1). The single 
equation <p"0/w=0 would, obviously be insufficient to determine the gravita- 
tional field- 



EINSTEIN'S LAW OF G-RAVITATION 43 

All laws, relating to phenomena in a geometrical field of 
force, which depend on the g's and their -first derivatives, will 
also hold in a permanent gravitational field. Laws which. 
depend oa the second derivatives of the g's will not necessarily 
apply. 

It must be remembered that we give no proof of this ; it 
is merely an explicit statement of our assumptions. It would 
be quite consistent with the general idea of relativity if the 
true expression of such laws involved the Riemann-ChristoSel 
tensor, which vanishes in the artificial field, and would have 
to be replaced before the equations were applied to the 
gravitational field. But if we were to admit that, the principle 
of equivalence would become absolutely useless. 

CHE GRAVITATIONAL FIELD OF A PARTICLE* 
28. We have seen that the gravitational-potentials satisfy 
the equations (26*3) 




. (28-1) 

We shall now find a solution of these equations corresponding 
to the field of a particle at rest at the origin of space-co- 
ordinates. We choose polar co-ordinates, vi#., 



In making this statement we are departing somewhat from 
the standpoint of general relativity. Strictly speaking, we can 
only define a system of co-ordinates by the form of the 
corresponding expression for ds*, that is by the gravitatioL- 
potentials. So that to ^specify the co-ordinates that are used 
involves solving the problem. Further, we have at present no 
knowledge of a particle of matter, except that it must be a 
point where the equations (28-1), which hold at points outside 
matter, break down ; we can only distinguish a particle from 
other mathematically possible singularities, such as doublets, 
by the symmetry of the resulting field. Thus the logical 
course is to find a solution, and afterwards discuss what 
distribution of matter and what system of co-ordinates it 
represents. We shall, however, find it more profitable to 
accept the guidance of our current approximate ideas in order 
to arrive at the required solution inductively. 



44 RELATIVITY THEORY OF GRAVITATION. 

The line-element ds can be assumed to be of the form 



e v dt* . (28-21) 

where h, // 3 v are functions of r only. 

The omission of the product terms, drdd, drd<p, dddy, is 
justified by the symmetry of polar co-ordinates ; the omission 
of drdt, d6dt, dydt involves the symmetry of a static field with 
respect to past and future time. If the latter products were 
present we should interpret the co-ordinates as changing with 
the time. 

A further simplification can be made by writing rV=r' 2 
and adopting r' as our new co-ordinate (dropping the accent). 
The resulting change in dr 2 is absorbed by taking a new L 
Thus the coefficient e? is made to disappear and we have 

ds*= -e x dr* -r W -r*sin 2 0dcp 2 +e v dt*. . (28-22) 

Comparing (28 22) with (15 3), we have 

0u=-^ 0r=-^ fe--r 2 sin 2 0, g u =e . (28-31) 
and SWO? wheno-=/=rT. 

The determinant g reduces to its leading diagonal, so that 

_ ? =^+>'r 4 sin 2 0, ..... (2832) 
and sr=l/9~ ....... (28-33) 

We can now calculate the three-index symbols (21-12) 



Since the ^'s vanish except when the two suffixes agree, 
the summation disappears and we have 



If <r, T, p are unequal we get the following possible cases : 



> . (28-4) 



{<rr, p}=0. 
None of the above expressions are to be summed. 



EINSTEIN'S LAW OF GRAVITATIOK 



45 



Whence by (28 31), denoting differentiation with respect to 
r by accents, we obtain 

{11, 1} =P' 
{12, 2}=l/r 
{l3, 3}=l/r 
il4, 4} =$*' 
^22, l}=-r- x } . (285) 

{23, 3} = cot e 

{33, 1} = -r sin 2 0<r x 
{33, 2} = - sin 6 cos 9 
{44, i} = i^-v 

The remaining 31 Christoffel symbols are zero. It should 
be noted that {21, 2} is the same as {12, 2} , etc. 

It is now not difficult to obtain the equations of the field. 
To assist the reader in carrying through the substitutions, we 
shall write out in full the equations (28-1) omitting the terms 
(223 in number), which obviously vanish. The following come 
respectively from G ll9 ? 22 , 6 ? 33 , <T 44 =0 : 

-f- {11, 4} + {11, 1} (11, 1} + {12, 2} {12, 2} + {13, 3} {13, 3} 
uf 



+ {14, 4} {14, 4} + ,-ilog V-g- ill, 1} r-log V -g=0 

(JT OT 

-I- {22, 1} +2 {22, 1} {12, 2} + {23, 3} {23, 3} +|dog V^g 

Qf 7t/ 

- {22, 1} ^log V~gr=0 
ff> 

-j- r {33, 1} -^ {33, 2} +2 {33, 1} {13, 3} +2 {33, 2} {23, 3} 

-- r {44, l}+2 [44, 1} {14, 4} - {44, 1} JUog V^g=0. 
Of the remaining equations, <? 12 =0 gives 
{13, 3} {23, 3} - {12, 2} ^ 

0" 

which disappears when the values of the symbols are substi- 
tuted ; and in the others there are no surviving terms. 

E 



4:6 EELATIVITY THEOBY OF GRAVITATION. 

Substituting from (28-5) and (28 32) the four equations give 
immediately 



_____ 

- _ -- -, 



*-2 cos 2 



These reduce to 

_t-iw,/_ i'\\ i ^=fk 

. (28-6) 



From tlie first and last equations /l / = /, and since both 
1 and v must tend to zero at infinity 1= v. The second and 
third equations (which are identical) then give 



Set e"=y, then 
Whence 

where 2m is a constant of integration, tn will later be identified 
with the mass of the particle in gravitational units. This 
solution satisfies the first and fourth equations, and, therefore, 
substituting in (28*22), we have as a possible expression for 
the line-element 

WW-~f 2 sin 2 8dy*+ydt*, . (28*8) 



=l 2m/r. 

It will be seen that the measured space around a particle is 
not Euclidean. Any actual measurement with our clock-scale 
gives the invariant quantity ds. If we lay our measuring-rod 
transversely, ds^rdO, so that our transverse measures are 



EINSTEIN'S LAW OF GBAVITATIQN. 47 

correct in this system of co-ordinates ; but if we lay it radially, 
dsy^dr, and the measures need to be multiplied by y* to 
give dr. Titus, referring our results to Euclidean space, we 
may say that a standard measuring rod contracts when turned 
from the transverse to the radial direction. 

We could, of course, decide to treat the radial measures as 
correct, and apply corrections to the transverse measures. 
This amounts to substituting &r f for y~*dr in (28-8), and using 
r' as the radial co-ordinate. It is impossible to say which 
form of (28-8) corresponds to our ordinary polar co-ordinates, 
since we have never hitherto had to pay attention to the 
ambiguity. 

The possibility of using any function of r, instead of r, for 
the distance is connected with the fact that Einstein's equations 
amount to only 6 independent relations between the 10 # ? s 
Consequently, quite apart from boundary conditions, there is 
a large amount of arbitrariness in choice of ^'s, ^.e., of co- 
ordinates. The reader may meet elsewhere with different 
expressions for the line-element due to a particle. The one 
adopted here was first given by Schwar^schild 

For some purposes the following analogy is helpful. Instead 
of considering continuous space-time, consider that funda- 
mentally we are dealing with an aggregate of points. With 
Galilean co-ordinates x, y, z, t\/ 1 the points are uniformly 
packed. Any measure that we make is really a counting of 
points, and a particle always moves so as to pass through the 
fewest possible points between any two positions on its path. 
Ajay mathematical transformation of these co-ordinates dis- 
turbs, without disordering, the distribution of the points in 
space ; but it is meaningless so long as we consider only the 
points and not the arbitrary continuous space we place them 
in. In a gravitational field the points are disordered according 
to some definite law. We can evidently re-arrange them so 
that the number of points in the circumference of a circle is 
less than n times the number in the diameter (a circle being a 
geodesic on a hypersphere, which is a locus such that the 
minimum number of points between any point on it and a 
fixed point called the centre is constant). 

This representation, however, gives only imaginary time and 
therefore imaginary motions. When extended to real motions 
it becomes too complex to be of much help. 



CRAPTEE 



THE CRUCIAL PHENOMENA. 

29. The Equations of Mot^on of a Particle m the Gravitational 
Field. 

dx 
Denote the contra variant vector ^ by A*. Then by (22-5 ) 

its covariant derivative is 



Multiply this by ^x a /Ss, we have 



showing that the right-hand side is a contravariant vector. 
Consider the equations 

^+{a/U}% 3 f=0, (T=l,2,8,4), . . (29) 

since the left-side is a vector, the equations will be satisfied 
(or not) independently of the choice of co-ordinates. In 
Galilean co-ordinates, the second term vanishes, and the 
equations reduce to ^a^/c^Q, which are the equations of a 
straight line. Equation (29) is thus the general equation oi 
the locus which in Galilean co-ordinates becomes a straight 
line. 

The path of a particle in Galilean co-ordinates (i.e., under no 
forces) is a straight line. The equations (29) are accordingly 
the equations of motion of a particle referred to any axes, 
provided there is no permanent gravitational field. Further, 
since they contain only first derivatives of the y's, in accord- 
ance with 27, these equations of motion will hold also when 
there is a permanent gravitational field* 

The equations must evidently correspond to the condition, 
fd$ is stationary, 



THE CRUCIAL PHENOMENA. 49 

and could have been deduced from it by tlie calculus of 
variations. The path of a particle is a geodesic in all cases. 
It should be noticed that /ds is not generally a mimmum. 

30. Using the values (28-5) of Christoffel's symbols, the 
equation of motion (29) for <r=2 becomes 

, 2 dr & A /OA io\ 

--- --rssO. . (3042) 
r ds ds 

Choose co-ordinates so that the particle moves initially in 
the plane 0=^/2 ; then d6fds=Q initially, and cos 0=0, 
so that d*6jds 2 ==Q. The particle therefore continues to move 
in this plane. The equations for cr=l 5 3, 4 are then 



=<' 18 > 

AH ,fa dt f 

j^-v' =0 (30 '14) 

ds^ ds ds v ; 

Integrating (30-13) and (30-14), we have 

r 2 ^-^ ........ (30-21) 



=ce -v ..... (30-22) 

ds y 

where h and c are constants of integration. 

Instead of troubling to integrate (30*11), we can use (28-8), 
which plays the part of an integral of energy, viz., 



From these three integrals, 
/eZn, 2 . 9 /(Z 

&}+*( 

or substituting for y its value (28-7) 



witk 



50 RELATIVITY THEORY OF GRAVITATION* 

Compare these with the ordinary Newtonian equations f of 
elliptic motion, 



I- (304) 



To make them correspond we must take c*=lm/a, 
where a is the major semiaxis of the orbit. The term 2mk*/r* 
represents a small additional effect not predicted by the 
Newtonian theory. Further, the quantity m, introduced as a 
constant of integration, is now identified as the mass of the 
attracting particle measured in gravitational units. With 
regard to the use of ds instead of dt in (30-3) 5 it must be 
remembered that ds is the " proper time " for the moving 
particle, so it is permissible to take ds as corresponding to the 
time in making a comparison with Newtonian dynamics. 

Mass, time and distance are all ambiguously defined in 
Newtonian dynamics, and in defining them for the present 
theory we have some freedom of choice, provided that our 
definition agrees with the Newtonian definition in the limiting 
case of a vanishing field of force. 

31. The Perihelion of Mercury. 

The ratio mja or mfr is very small in all practical applica- 
tions. If we take 1 kilometre as the unit of length and time 

\ nOOQQQ sec * )' ^ en * or *k e earth's orbit a=149. 10*, and 

the angular velocity o>=6*64: . 10~ 13 . Hence the mass of the 
sun, 

w=eo 2 a 3 = 147 kilometres. . . . (31-1) 

Thus for applications in the solar system m/r is of order 10~ 8 
and it is easily seen that A 2 /r 3 is of the same order. Also the 
difference between dt and ds is of order 10""%. 

From (30-3) we have 

(L J^*+*=& 

V 3 d<?) ^r* (C 
or writing u=l/r. 



THE CRUCIAL PHENOMENA. 51 

Differentiating with respect to q>, 

; .... (312) 



Since A 2 u 2 is of order 10" 8 we obtain an approximate solution 
by neglecting 3wu 2 . This is 

t*=5(l+eooB(9-tar)), . . . (31-3) 
ti 

as in Newtonian dynamics. 

For a second approximation, we substitute this value of u 
in tie small term Smu 2 , and (31 2) becomes 

w , 3m 3 . 6m 3 



(314) 

Of tlie small additional terms tte only one wHch. can gwe 
appreciable effects is the term in cos (9^), wticli is of the 
proper period to produce a continually increasing effect by 
resonance. It is well known that the particular integral of 



is 

u~^A<p sin 9. 

Hence this term gives a part of n, 



Adding this to (31-3) we have 

cos (9-cr)+-9 6 sin 



where to=9, and (5tnr) 2 is neglected. 
ti> 

Thus whilst the planet moves through one revolution, the 
perihelion advances a fraction of a revolution equal to 

dT3 3m 2 3m _ 



where T is the period of the planet, and the velocity o lighi & 
has been re-instated. 



52 RELATIVITY THEORY OF GRAVITATION. 

For the four inner planets the numerical values of this 
predicted motion of the perihelion are (per century) : 



Mercury +42"-9 +S"-82 

Venus 8-6 005 

Earth 3-8 0-07 

Mars 135 0-13 

The value of ed is given because this corresponds to the 
perturbation which can be measured. Clearly when e is 
vanishingly small it is not possible to detect observationally 
any change in the position of perihelion. The orbits of Venus 
and the Earth are nearly circular so that the predicted effect 
is too small to detect. 

The following table gives the outstanding discrepancies 
between the present theory and observation for efcr and de 
(per century) with their probable errors. The seculai changes 
8e are analogous to edtt ; and the two perturbations may be 
regarded as the two rectangular components of a vector. In 
the last column we give the outstanding discrepancies of edtz 
on the Newtonian theory ; those of de are, of course, unaltered. 

Einstein's Theory. Newtonian. 
eto de edrz 

Mercury -<T-58 ilT-29 ~0"-88 0-"33 +S"24 

Venus -Oil 017 +0-21 0-21 -0 06 

Earth 0-00 0-09 +0-02 0-07 +007 

Mars +051+023 +029 0-18 +O64 

It will be seen that the famous large discordance of the 
perihelion of Mercury is removed by Einstein's theory. No 
other charge of importance is made except a slight improve- 
ment, for the perihelion of Mars. Of the eight residuals, four 
exceed the probable error, and none exceed three times the 
probable error, so that the agreement is very satisfactory. 

It may be noticed that according to (314) the orbit is" not 
exactly an ellipse, even apart from this progression of the apse. 
But this (unlike the motion of perihelion) has no observational 
significance, and merely arises from our particular choice of 
measurement of r. In any case the curve in non-Euclidean 
space, which is to be described as an ellipse, must be a matter 
o convention. 



THE CRUCIAL PHENOMENA. 53 

It will be found (putting dr/ds~Q in (30-11)) that for a 
circular orbit Kepler's third law is exactly fulfilled. This 
again is not an observable fact. To compare it with obser- 
vation we should have to consider the nature of the astro- 
nomical observations from which the direct value of the axis 
of the orbit is measured. 

32. Deflection of a Ray of Light. 

In the absence of a gravitational field the velocity of light 
is unity, so that 

dx\* /<%\ 2 , fdz \ 2 __ 

dt) + \dt) + \dt) - 1 ' 



Accordingly ds*=~-dx z -dy*-dz*+dt 2 ^Q. . . (32-1) 

Hence for the motion of light <fo=0, and by the principle of 
equivalence this invariant equation must hold also in the 
gravitational field, 

It may be of interest to not.ce that for an observer travelling 
with the light, dx=dy=^dz=0, so that <&=<Zs=0. Hence, if 
man wishes to achieve immortality and eternal youth, all he 
has to do is to cruise about space with the velocity of light, 
He will return to the earth after what seems to him an instant 
to find many centuries passed away. 

Setting efo=0 in (28 8) we have (for motion in a plane) 



Hence if v is the velocity of light in a direction making an 
angle F with the radius vector, 

v*(y l cos 2 F+sin 2 F)=y, 
whence *=y(l -(1-y) sin 2 F)-* . . . , (32-3) 

The velocity thus depends on the direction ; but it must 
be remembered that this co-ordinate velocity is not the velocity 
found directly from measures at the point considered. When 
we determine the velocity by measures made in a small region, 
and use natural measure ^.e., g^ having the values (16-3) at 
that point), the measured velocity is necessarily unity. 

Since it is inconvenient to have the velocity of light varying 
with direction, we shall slightly alter our co-ordinates. Set 

r=r^m ....... (324) 

Then, neglecting squares of 



54 RELATIVITY THEORY OE GBAVITATIOJtf. 

Substituting in (32 2) 



so that in these co-ordinates, 

t;=y=l-2w/r-fcl-2w/ri . . . (32-5) 
i or all directions. We can now drop the suffix of r x . 

By Huygens' principle the direction of the ray is determined 
by the condition that the time between two points is stationary 
for small variations of the path. The course of the ray will 
therefore depend only on the variation of velocity, and will be 
the same as in a Euclidean space filled with material of suitable 
refractive index. The necessary refractive index ju is given by 



We thus see that the gravitational field round a particle will 
act like a converging lens. 
The path of a ray through a medium stratified in concentric 

spheres is given by , 

* & J /p=const ....... (32-71) 

where p is the perpendicular from the centre on the tangent. 
By (32-6) we have to this order of approximation, 



^=1+ ....... (32-72) 

But (32*71) and (32*72) are the integrals of angular momen- 
tum and energy for the Newtonian motion of a particle with 
velocity p under the attraction of a mass 2m, the orbit being 
a hyperbola of semi-axis 2m. This hyperbola, therefore, gives 
the path of the light. If the distance from the focus to the 

apse is R, we have 
r 



., , , R R 

so that e=l+r~-=Q= - , 

2m Zm 

and the very small angle between the asymptotes 

2 2 4m 



Thus a ray of light travelling from QO to + oo 9 and passing 
at a distance R from a particle of mass m experiences a total 
deflection. . 



a= (32*8) 



"R 



THE CRUCIAL PHENOMENA. 



55 



For a star seen close to the limb of the sun, by (31 1) 
m~l 47 kin. ., and J?= sun's radius =697,000 km. Hence 

a-r-74. 

It is curious to notice the occurrence of the factor 2 (mass= 
2m) in the dynamical analogy. The deflection is twice what 
we should obtain on the Newtonian theory for a particle 
moving through the gravitational field with the velocity of 
light. The path of a light ray is not a geodesic (or rather 
the notion of a geodesic fails for mot on with the speed of 
aght) , it s detemrned by stationary values of fdt instead 
of fds. 

It may also be noted that the velocity of light decreases as 
the light f a 7 s to the attracting body. 




O 



FIG. 4. 



33. It is hoped to test this prediction by observations of 
stars near the limb of the sun during a total eclipse. If the 
answer should be in the affirmative., the question will arise 
whether this must be considered to confirm Einstein's law of 
gravitation, or whether the deflection is sufficiently accounted 
for by the simple hypothesis that the mass of the electro- 
magnetic energy of light is subject to gravitation. The 
unexpected factor 2 suggests that the deflection on Einstein's 
theory will be double that which would result from the ordinary 
electromagnetic theory. It is worth while to examine this 
more closely. 

Consider a tube of light of unit cross-section and length ds 
(Fig. 4). Let the inclination of the ray to the axis of x be y- 
Let g be the acceleration of the gravitational field directed 
along Oy. Let E be the energy per unit volume ; and c be 
the velocity of light, which on the electromagnetic theory ie 
absolutely constant. 



56 RELATIVITY THEORY OF GEAVITATIOIST. 

Then the mass of electromagnetic energy E, according to 
electromagnetic theory (or by (7-85)), is E/c*, so that if this 
is subject to gravity the momentum generated in the tube in 
unit time will be 



P 

~ 2 ds.g along Oy. 

c 



If the light is stopped by an absorbing screen placed 
perpendicular to the ray the radiation-pressure is numerically 
equal to E, showing that momentum E in the direction of 
the ray passes across a section of the tube in unit time. ^Thus, 
resolving in the a? and y directions, the conservation of 
momentum gives 



~5"(E cos ty) . ^$=0, 

(L Cfj& 

-=r(E sin w) . d$=^~ds, 

ds^ ^ c* 

dE rt dw 

Whence ^-cos w E sin w -~~ =0, 

ds ds 

. n *yj$ 



(334) 



Eliminating dE/ds , 

jf-Si* (33 ' 3) 

The radius of curvature ds/dy is thus c*Jg cos y, which is 
exactly the same as for a material particle moving with 
velocity c in ordinary dynamics. This, as shown in the last 
paragraph, is only half the deflection indicated by Einstein's 
theory ; and the experimental amount of the deflection should 
thus provide a crucial test. 

34. Displacement of Spectral Lines. 

Consider an atom vibrating at any point of the gravitational 
field. It is a natural clock which ought to give an invariant 
measure of an interval ds ; that is to say, the interval <5$ 
corresponding to one vibration of the atom is always the same, 
Let the atom be momentarily at rest in our system of co- 
ordinates (though subject to the acceleration of the field) ; 
then dx^dy~dz~Q, and by (15-3) 



THE CRUCIAL PHENOMENA. 57 

If then dt and At' are the periods of two simiar atoms 
v brating at different parts of the field where the potentials 
are g u and g\& respectively, 

dt r ...... (344) 



K t refers to an atom vibrating in the photosphere of the sun* 



_ 2m 
I 



and if If refers to an atom m a terrestrial laboratory, where 
gr' 44 is practically unity, 



/! ~ =1-00000212 . , . (34-2) 
dt -a 

The solar atom thus vibrates more slowly, and its spectral 
lines will be displaced towards the red. The amount is 
equivalent to the Doppler displacement due to a velocity of 
0-00000212, or in ordinary units 0-634 km. per sec. In the 
part of the spectrum usually investigated the displacement is 
about 0-008 tenth-metres. 

The effect is of particular importance, because it has been 
claimed that the existence of this displacement is disproved by 
observations of the solar spectrum.* The difficulties of the 
test are so great that we may perhaps suspend judgment ; 
but it would be idle to deny the seriousness of this apparent 
break-down of Einstein's theory. We shall therefore consider 
the phenomenon from a more elementary point of view. 

The phenomenon does not depend on the greater intensity 
of the field on the sun, but on the potential ; and ^it can 
evidently occur in a uniform gravitational field. Consider an 
observer in a uniform field of intensity g and two similar 
atoms A i and A^ A^ being close to the observer and A z at a 
distance a measured parallel to the field. The observer and 
his atoms will, of course, be falling with the Acceleration g. 
Consider them all enclosed in a room which is also falling ; 
then by the principle of equivalence cannot detect any 
effect of the field, and he will therefore observe the same 
period of vibration T for both atoms. Now refer the pheno- 
mena to unaccelerated axes which coincide with the accelerated 
axes at the instant $=0. The vibration emitted by A* at 
the time *=0 will reach at the time =a (the velocity of 

* C. E. St. John, " Astropfcysical Journal," Vol. 46, p. 249. 



58 RELATIVITY THEORY OF GRAVITATION* 

light being unity), by which t me will iiave acquired a velocity 
ga relative to tlie unaccelerated axes. He will, therefore, 
correct Ms observation of the period of A 2 for tlie Doppler 
effect of this velocity and deduce a true period T/(lga) 
The period of A mil require no correction, and wil still be 
given, as T. Since ga is the difference of potential between A l 
and A % this agrees with (34 2) 

As an example of a varying field, consider an observer O 
at the origin of co-ordinates and an atom A at a distance r 
in a field of centrifugal force of potential O^JcoV 2 , the atom 
being at rest at the time of emission of the light, but subject 
to the acceleration of the field. Another way of stating the 
problem is that there is no field of force, and the atom is 
moving with velocity cor at right angles to the radius vector 
at the time of emission of the light. But in that case the 
period of vibration is by 4 increased in the ratio 



by (16-2), 

as compared with the stationary atom. This again agrees 
with (34-1). 

These verifications seem to leave little chance of evading 
the conclusion that a displacement of the Fraunhof er lines is a 
necessary and fundamental condition for the acceptance of 
Einstein's theory ; and that if it is really non-existent, under 
conditions which strictly accord with those here postulated, 
we should have to reject the whole theory constructed on the 
principle of equivalence. Possibly a compromise might be 
effected by supposing that gravitation is an attribute only of 
matter in bulk and not of individual atoms ; but this would 
involve a fundamental restatement of the whole theory, v 

If the displacement of the solar lines were confirmed, it 
would be the first experimental evidence that relativity holds 
for quantum phenomena. 



CHAPTEK VI. 



THE GRAVITATION OF A CONTINUOUS 
DISTRIBUTION OF MATTER. 

35. la the problems occurring in Nature our data give, not 
the distribution of the individual atoms, but the large-scale 
average distribution of density. This transition from discrete 
particles to the equivalent continuous medium occurs in the 
Newtonian theory of attractions, and involves the replacement 
of Laplace's equation y 2 ?^ by Poisson's equation y 2 ?= 
4:7* p. We shall now find the corresponding modification of 
Einstein's equations 6v r =0. 

The equations 6v T =0 are not linear in the 0's, and conse- 
quently the fields of two or more particles are not strictly 
additive. But the deviations produced in the gr's by any 
natural gravitational field are extremely small, so we shall 
neglect the product terms and treat the fields as superposable. 
It will be shown below that ultimately this approximation 
does not produce any inaccuracy in the application we have in 
view. 

As in (324) we shall write r^r^m in (28-8) and neglect 
(m/r) 2 . Then the line element in the field surrounding the 
particle is 

- (35-1) 



We consider r x to be the actual radius vector, since the 
mode of measurement is arbitrary to this extent. Converting 
Into rectangular co-ordinates, 



. (35-2) 

The origin is now arbitrary, and r denotes the distance oi 
the attracting particle from the element ds. The effects of 
a number of particles being additive to our order of approxi- 



60 RELATIVITY THEORY OF GRAVITATION. 

mation, we shall have for any number of particles at rest 
relative to the axes, 

da*=-(l+2Q)(daP+dy^te*)+(l-2Q)dt* . (35-3) 

where O=Z(m/r)=the Newtonian potential. 

Consider a point in the medium where the density is p, 
and with as centre describe an infinitely small sphere. If 
we neglect the material inside the sphere, the equations of the 
gravitational field in free space will be satisfied at 0, i.e., 
6v T =0. Hence in calculating the values of 6> T at we need 
only take account of the material inside the sphere. Accord- 
ingly in (35-3) O refers to the potential inside an infinitely 
small sphere of uniform density p 

Since 9Q/9&, &c., vanish at 0, we have only to take account 
of terms in (28-1) containing second derivatives of the #'s ; 
and the calculation of & at is quite simple. We have 



*^'V ^ ' s /~)/y * ~ i 'J/7 
}X^ OX 2 0# 3 

22,7 f}2/ ^2/7 

--^logV-^ . (354) 
omitting 33 terms which vanish or cancel. 

A 1 J^\ _ -jj^ ^_ - "1 -y -^ ^ ^ ___ "I / Q K K \ 

and by '(35-3) 




Hence substituting in (354) 



= 4jcp, by Poisson's equation. 

Working out the other components similarly (with slight 
variations in the case of 6r 44 ) we find 



11=22 = ^33 = ^44= ~^p. . - (35-6) 

The scalar G^g^G^ 

^ -011 ~0 M -083+04* 

=87ip .......... (35.7) 

Now form the covariant tensor 

G^-teitf ..... (35-8) 



<3 R A VITATION OF A CONTINUO US DISTRIBUTION OP MATTER. 61 

We have by (35-6) and (35-7) 

2>=p, 

and all other components vanish.. 

Having thus found the value of T^ in this special system of 
co-ordiuates we could find its general value by (19*31). It is, 
however, simpler to proceed as follows. It x^ is a co-ordinate 
of a point in the material, consider the quantity, 

pt"-^ ....... (3591) 

* ds ds v ' 

Since with respect to our special axes the material is at rest, 
=0 (,1=1,2,3), and ^=1 (/K =4). 

ICo too 

Hence all the components of (35-91) vanish except for 
j M =j;=4 J for which the component is p just like T ffr . This, 
however, is a contravariant tensor * and (35-8) requires a 
covariant tensor. 

We therefore form the associated covariant tensor ( 



. (35-92) 



which agrees with (35*91) in our special co-ordinates. 

The equations (35-8) and (35-92) are in covariant form, and 
are true in one system, hence they are true in all possible 
systems of co-ordinates. They are the general equations of 
the gravitational field in a continuous medium. 

An alternative form of (35-8) is readily obtained, viz., 



-% g<rr T), . . . (35-93 

where T is the associated scalar <7 < "T aT . (This follows since on 
inner multiplication of (35-8) by g we obtain Gr=^SnT.) 

36. We thus find that in a continuous medium, G- ffr , instead 
of vanishing, is equal to a tensor expressing the content and 
state of motion of the medium at the point considered. On 
the equations here found we have two observations to make. 

(1 ) A little consideration will show that notwithstanding the 
approximations made at various stages of the proof, the results 
are quite rigorous. It is clear that so far as the calculations 
for the infinitely small sphere surrounding O are concerned, 

* p is to be treated as an invariant. Whatever the axes chosen, /> is to 
Ve the density in natural measure as estimated by an observer moving with 
the matter. 

F 



62 RELATIVITY THEOKY OF GRAVITATION. 

we are justified in neglecting the product terms, since in the 
limit they will vanish compared with the linear terms. 
Another way of seeing this is to consider that G ffr involves 
only derivatives up to the second at the origin ; and there- 
fore we need only expand the g's in powers of r as far as r 2 ; 
but in our units p is of dimensions r" 2 , and since the g"s in 
rectangular co-ordinates are of zero dimensions, any terms 
involving p 2 would be of the form p 2 r 4 , and therefore need 
not be retained. The effect of the gravitation of the matter 
outside the sphere is eliminated completely by our choice of 
co-ordinates. We chose them so that at the # 5 s have the 
values (16 3) ? i.e., we use " natural measure.' 5 Since our axes 
move with the matter at 0, the first derivatives of the 0*s 
(expressing the force) will not vanish unless the matter at O 
is moving with the acceleration of the field, which is not the 
case if there is any internal stress These first derivatives 
are omitted from our equations after (35-3), because as already 
explained the external matter alone contributes nothing to 
G<rr\ further, the cross-terms are zero, because the first de- 
rivatives of the #'s arising from the matter inside the sphere 
vanish. The result is thus rigorous, provided that in measuring 
the invariant density p we use natural measure, ^.e., the mass 
and unit volume must be taken according to the direct 
measures made by an observer at moving with the material 
there. 

The argument may be summarised thus : 6> T consists of 
terms of types 

/ 2 +j 2 +JiH-^i#i+#iH-terms in I +terms in # , 
where / and E refer to the matter internal and external to 
the small sphere, and the suffixes refer to the order of the 
derivatives. Terms in I x vanish by the symmetry of the 
sphere ; terms in 7 vanish as the sphere is made infinitely 
small ; terms in E vanish because we use natural measure ; 
the terms E^E^ vanish by Einstein's equations for free 
space. All that is left is J 2 , and as the sphere is made infinitely 
small our determination of its value becomes rigorous. 

(2) In replacing a molecular medium by a continuous 
medium, it is not sufficient to average the distribution of mass 
and mass-motion only ; we must also represent somehow the 
internal motions. This is done by adding another property to 
the continuous medium the pressure, or stress-system. The 
tensor T vr will contain terms corresponding to the pressure ; 
these are negligible in practical calculations of the gravitational 



GKAVITATION OP A CONTINUOUS DISTRIBUTION OF MATTER. 63 

field because the pressure is of order p times the Newtonian 
potential, i.e., of order p a . The terms are, however, 
important in the general equations of momentum and 
energy, and we shall consider them more fully in the next 
paragraph. 

37. In the dynamics of a continuous medium the most 
fundamental part is taken by the associated mixed tensor, 

T;=rt a =^ Po ^^, . . . (37-1) 

where we have inserted the 2 in order to take account of the 
variety of internal motions, and have written p for p in order 
to call attention to the fact that it represents the density in 
natural measure and not the density referred to the arbitrary 
axes chosen. 

T may be called the energy-tensor, though it is actually an 
omnmm gatherum of energy, mass, stress and momentum. 

First consider the meaning of this tensor in the absence of a 
gravitational field, and accordingly choose Galilean axes. If 
u, v, w are the component velocities of the particles, 

' (37.2) 



But by (? 92) the density referred to the axes chosen is 



Hence T^g^X ...... (37-3) 



Putting in the Galilean values of g^, we have 

l*=-~ JSpw 2 , Zpvw, Zgwu, 
^^ -2; pW j, -Zpu 2 , -Z$wv, Zpv (374) 

ZpwU'', Zpvw, Zpw 2 , Zpw 



This tensor may be separated into two parts, the first referring 
to the motion, w , t? , W Q , of the centre of mass of the particles 
in an element, and the second to their internal motions, 
tf i *>!> w i> relative to the centre of mass. With regard to the 
last part, 2^u 1 v l represents the rate of transfer of 
M-momentum across unit area parallel to the y-plane, and is 



64 RELATIVITY THEORY OF GRAVITATION. 

therefore equal to the stress usually denoted by p x , . Hence 
(374) becomes 




p 
(37-5) 

where p is now the whole density referred to the axes chosen. 
Consider the equations 

-iz^O ........ (37*6) 

*v * 

Taking ^=4, and using (37*5), we get the well-known 
equation of continuity 



- ... 

ox 
Taking /*=!, 




Now (37-7) and (37 8) are the fundamental equations of 
hydrodynamics. By assuming Galilean axes we have neglected 
any extraneous body-forces, and so the term pX, which 
occurs on the right side of (37-8) in the more general form of 
the equation, does not appear in this case. 

The equation (37-6) is thus equivalent to the general equa- 
tions of a fluid under no forces. 

31 The equat-n 3TJ/3 ^=0 represents a law of con- 
servation. hoose one of the co-o dinates, a? 4 , as independent 
variab'e, and integrate the equation through a three-dimen- 
sional volume marked out in the other co-ordinates. This gives 

*** 

=the surface integral of the normal con * 
pone t of (Tl, 2% JJ). 

If the volume is such as to include the whole of the material, 
7* vanishes on the surface ; the surface-integral therefore 



GRAVITATION OF A CONTINUOUS DISTRIBUTION OF MATTER. 65 

vanishes, and hence the volume integral of T* remains constant. 
If the surface does not include all the matter, any change of its 
content of T * occurs by a flux across the surface measured by 
(2% % jTJ). It will be seen from (37-5) that for the axes there 
used T^ represents the negative momentum and the mass (or 
energy), and that T &c., represent the flux of these quantities. 
Equation (37-6) therefore gives the law of conservation o 
momentum and mass, as may be verified from the correspond- 
in 5 hydromechanical equations. 

39. Equation (37*6) is the degenerate form for Galilean co- 
ordinates of the co variant equation 

5^=0 ....... (39-11) 

where T, is the (contracted) covariant derivative of ZJ (tee 
(22 -7)). Equation (39- 1 1 ) thus holds for Galilean co-ordinates, 
and it does not contain derivatives of the gr's higher than the 
first. Hence by the principle of equivalence it holds generally, 
inc uding the case of a permanent gravitational field. 

Taking equation (35-8) 



multiply by g rv . We obtain 

G-^=-&*Z .... (39-12) 
Take the covariant derivative of both sides, and contract it, 



. . . (39-13) 
whence by (20-1) 



Clearly this equation will have to be an identity, and it may 
be verified analytically, using the values (26-3) of G^. For 
cr1, 2, 3, 4, this identity gives the four relations between 
Einstein's ten equations, which have already been mentioned 
as reducing the number of independent conditions to six. 

Conversely, from the identity (39-14) we can deduce (39-11), 
and hence obtain the equations of hydromechanics and the 
law of conservation directly from Einstein's law of gravitation. 
Further, by applying the hydromechanical equations to an 
isolated particle., we obtain the equations of motion (29). 
The mass of a particle has been introduced first as a constant 
of integration, and afterwards identified with the gravitation- 
mass by determining the motion of a particle in its field ; it 
now appears that it is also the inertia-mass, because it satisfies 



66 RELATIVITY THEORY OF GRAVITATION, 

the law of conservation of mass and momentum, which gives 
the recognised definition of inertia. 

It is startling to find that the whole of the dynamics of 
material systems is contained in the law of gravitation ; at 
first sight gravitation seems scarcely relevant in much of our 
dynamics. But there is a natural explanation. A particle of 
matter is a singularity in the gravitational field, and its mass 
is the pole-strength of the singularity ; consequently the laws 
of motion of the singularities must be contained in the field- 
equations, just as those of electromagnetic singularities (elec~ 
trons) are contained in the electromagnetic field-equations. 
The fact that Einstein's law predicts these well-known pro- 
perties of matter seems to be a valuable confirmation of this 
theory, 

The general equation (39-11) enables us to pass from the 
equations of a fluid under no body forces to the equations of a 
fluid in a field of force. It can be simplified considerably. By 
(22-7) 



By (26-25) the last term becomes 



The second term is equal to 



= _ 

2 a* <** 

since the other two terms cancel on summation, 
1 



' a 



This last result follows, since 

9^=0 or 1, 

so that rtu+^ e % va =0, 

Multiply by g* and use (20-15), we obtain 

ff^g-dg^-dg* ...... (394) 

Hence inserting (39-22) and (39-3) in (39 21), we have 



GRAVITATION OF A CONTINUOUS DISTRIBUTION OF MATTER. 67 

This equation has its simplest interpretation when we choose 
co-ordinates, so that V g=l, that is to say, the volume of a 
four-dimensional element is to be the same in co-ordinate 
measure as in natural measure. Owing to the considerable 
freedom of choice of co-ordinates, allowed by Einstein's equa- 
tions, it is always possible to do this. In that case (3945) 
becomes 



Comparing this with (37 6), which holds when there is no 
field of force, we see that the term on the right represents 
the momentum and energy transferred from the gravitational 
field to the material system. As a first approximation (re- 
taining only T 44 =p, and g 44 =l 2Q) we see that it gives, 
for ju=l, 2, 3, the terms pJf, pY, pZ of the usual hydro- 
dynamical equations, which were omitted in (37 8). 

40. Propagation of Gravitation. 

The velocity of light being a fundamental relation between 
the measures of time and space, we may expect the strains 
representing a varying gravitational field to be propagated 
with this velocity. We shall show how to derive the equations 
exhibiting the propagation. 

In the theory of sound, the general equation of disturbances 
propagated with unit velocity is 



where <D is zero except at the source of the disturbance. The 
general solution is 



the integral being taken through the volume occupied by the 
source of disturbance, and the value of ' taken for a time 
I _ r ' 3 where r' is the distance of the volume dV from the point 
considered. Thus 9 is a retarded potential, and (40-12) ex- 
hibits the effect as delayed by propagation. 

In the case of sound the velocity depends to a slight extent 
on the amplitude, and (40-11) is only strictly true if the square 
of 9 is negligible. Similarly the velocity of light depends to a 
slight extent on the gravitational field ( 32) ; consequently we 



68 RELATIVITY THEORY Off GRAVITATION, 

can only expect to obtain an equation of this form if^ w 
neglect the square of the disturbance, so that the equation 
become linear. 

The origin of gravitational waves must be attributed^ 
moving matter ; and, since G> vanishes except in a regioi 
occupied by matter, we may take G^ as the analogue of <D 
We shall examine whether the disturbance can be representec 
by a quantity h^ satisfying 

=2GU, ..... (4021 



where the exact significance of h^ is yet to be found. We 
shall regard h^ as a small quantity of the first order ; the 
deviations of the g^ from their Galilean values will also be of 
the first order. Small quantities of the second order will be 
neglected. 
If, as usual, 



and A 

Then, multiplying (40 21) successively by g v " and g* v , we have 
to this approximation,* 

DA*=2SJ . . . . . (40-22) 
and Q&=2 ..... (40-23) 

Hence Q(*J -&#) =2(6? -Jtf0) 

{=z~lnTl by (39-12). 

To the present approximation (37-6) holds, so that 



Having regard to boundary conditions, the solution is clearly 






* The gp behave as constants until we reach equation (40-5), because 
their derivatives, which are small quantities of the first order, only appear 
in combination with the small quantities k^y or G/w. The gp>v accordingly 
pass freely under the differential operators. 



GRAVITATION OF A CONTINUOUS DISTRIBUTION O F MAT f ER. 69 

Consider the expression 
_1 A. j^3*tf .8^6. 

2 aB.t ff Va^^a^ a 

which to our approximation 



By (40-3) the first two terms cancel with the last, and for 
Galilean values of g*^ the third term is simply 



Thus by (40-21) the expression (404) reduces to G>. 

Neglecting squares of small quantities, G> (26-3) reduces to 



2 a 

Comparing (404) and (40-5) we see that the A's must be 
equal to the gr's or rather since the A's have been treated as 
small quantities, they must be the deviations of the #'s from 
their constant Galilean values. Writing d^ v for the Galilean 
values of g^ v (16 3), then 

fc=<5^+7w, ..... (40-6) 
and A^ satisfies the equation of wave-propagation (40*21). 

By (40*12) the solution of the propagation equation is 



This can be used for the practical calculation of g^ due to 
an arbitrary distribution of moving matter. It is necessary, 
as in the corresponding calculation of retarded electromagnetic 
potentials, to allow for the variation of // from point to 
point of the body ; the boundary of &V does not coincide with 
the limits of the body at any one instant. Thus for a particle 
of mass m, we have * 



* See, for example, Lorentz, " The Theory of Electrons," p. 254 ; or 
nnragligutti, " The Principle oi Bel&tivity," p 108. 



Cunnragligutti, 



70 RELATIVITY THEORY OF GRAVITATION. 

where v r is the velocity in the direction of r, and the square 
bracket indicates retarded values. As is well known [r(l 0,)] 
is to the first order equal to the unretarded distance r, so that 
notwithstanding the finite velocity of propagation the force is 
directed approximately towards the contemporaneous position 
oi the attracting body. It was lack of knowledge of this 
compensation which led Laplace and many following him to 
state that the velocity of gravitation must far exceed the 
velocity of light. 

The practical application of these formulae is, however, very 
limited. In a natural system (e.g , the solar system) the 
relative velocities (u) are due to the gravitational field and u* 
is a small quantity of the first order. Consequently our 
approximation is not good enough to take account of T ll7 T 12 , 
&c., in natural systems ; it can only include components with 
suffix 4.* The fact is that the whole idea of propagation from 
a point-source is an abstraction ; actually the motion of the 
source, or singularity, is but the symbol of the changes occurring 
in all parts of the field ; we cannot say whether the motion is 
the cause or effect of the gravitational waves. 

The present solution is a particular solution. It gives 
unique values of the ^ , but these may, of course, be subjected 
to arbitrary transformations. 



* For the higher approximations needed in the problem* of the solar 
system, see De Sitter, " Monthly Notices," Dec. 1916. 



CHAPTER VII. 



THE PRINCIPLE OF LEAST ACTION. 

41. Lagrange's Equat^ons. 

We shall again restrict the choice of co-ordinates so that 
V __ i. Einstein's equations (26-3) for the field in free space 
then becomes simplified to 

^=--i-{^o}+{^,a}{-a,^}=0. . (41-1) 

O&a. 

We shall regard g^ v as a generalised co-ordinate (g), and 
a? l3 x 2 , #3, #4 as independent variables a four-dimensional time. 
Writing g% v for fig^/dx^, which will then be a generalised 

velocity (q), we shall show that equations (414) can be ex- 
pressed in the Lagrangian form. 



where Z=gr {/ift a} {vo, /?} ..... (41-8) 

it being understood that the g^ are expressed as functions of 
the gr. 

We have from (41-3) 

L {& a} [va, $} dg v +2g>" (^ a] d {va, 0} , 
since in the last term /* and v are dummies. 

= - f^, a} 
But 



The last two terms in the bracket will cancel in the summa- 
tion after inner multiplication by {/*j8, a} , because ^ and /#, 
v and A are interchangeable -simultaneously. Also by (394) 



Hence di= - {/i/J, a} (va, 



72 RELATIVITY THEORY OF 

Therefore 



showing that (41-1) and (41-2) are equivalent. 

As in ordinary dynamics, Lagrange's equations are equiva- 
lent to 

is stationary .... (41*5) 



iLdz 



for variations of g*", dr being the iour-dimensional element of 
volume, here representing the independent variable. It must 
be remembered that the variations are limited by the con- 
straint V <7=1. 

42. Principle of Least Action.* 

Following out the dynamical analogy dL/dg* v or dLfiq 
is to be regarded as a momentum (p). The system is dynamic- 
ally of the simplest kind, since L does not contain the <f time," 
XP, explicitly, and ;t is a homogeneous quadratic function of 
the " velocities." By the properties of homogeneous functions 

.at 

2L=Zq-Z 

v" 

Since (pq+qp) is a perfect differential, 
]S(pq+qp}dT 

will be equal to a surface integral ; and it will, therefore be 
stationary for variations of g* v (the variations as usua being 
supposed to vanish at the boundary). 

Thus 8f2qpdT=-dl2qpdr=-2dJLdr . . (424) 

Hence, if we write 

H=*L+2qp ...... (42-2) 

by (41-5) and (42-1) 

Iffdr is stationary. . f . (42-3) 

* The strict analogue of the principle of least action is the stationary 
property of fj^qyd^ The restriction in dynamics that the energy is not 
to be varied corresponds to V g=\ t (Cf. 43) 



THE PRINCIPLE OF LEAST ACTION. 73 

By (414) 



Hence (42-2), (41-3) and (41-1) give 



We can therefore write the result (42-3) thus 

J G . V--J. & is stationary . * (42 4) 



J 



since V </=l. _ 

But 6r and V g . dr are invariants (20-3) ; so that (42-4) has 
no reference to any particular choice of co-ordinates, and the 
restriction V^==l can now be removed. It is thus a more 
general result than (41-5). 

43. Energy of the Gravitational Field. 

Reverting to the restriction V 0=1, multiply (41-2) by tf 



. 

Remembering that 



we have, adding (43-1) and (4 -2), 

3-^\ 8-" i A ^ Q\ 

v ,} .... (*0'd) 



= -IfajLr f , (43-4) 

where -16rc S=ffS'^-ffSl- (* 3 ' 5 ) 



We have used the property of g$ as a substitution operator. 
The quantity t$ defined by (43-5) is the analogue of the 

^ T 
Hamiltonian integral of energy, 2qrL. In free space 

6^=0, and (434) becomes 

s ( ;- (4S - 6) 

showing that $ is conserved (38). 



74 RELATIVITY THEORY OF GRAVITATION. 

When matter is present (434) gives 



=== dx~ v ~~ 2 ^ v 

since, when g= 1, 
Hence by (35-8) 



= --r- by (39-5). 

d*'* 

Therefore 

0. ..... (43-8) 



This is the law of conservation in the general case when 
there is interaction between matter and the gravitational 
field. We see that the changes of energy and momentum of 
the matter can be regarded as due to a transfer from or to the 
gravitational field, the total amount being conserved. We 
have, in fact, traced the disappearing portion of the material 
tensor T^ and shown that it reappears as the quantity 
t belonging to the gravitational field. 

In order to represent the phenomena in this way we have 
had to restrict the choice of co-ordinates by keeping the volume 
of a region of space-time invariant (V g=l). Otherwise 
the equation takes the more general form (39-11) which cannot 
immediately be interpreted as a law of conservation. It should 
be noted that, unlike Tp, the quantity is not strictly a tensor. 

44:. The Method of Hilfoert and Lorentz. 

An alternative method of deriving the fundamental equations 
of this theory is based on the postulate that all the laws of 
mechanics can be summed up in a generalised principle of 
stationary action, viz., 



....)\.dT=Q.. . (44-1) 

Here H^H^H^Bxe invariants* involving, respectively, the 
parameters describing the gravitational field, the electro- 
magnetic field, and the material system. If we consider 

* Invariant because the equation must hold in all systems of co~ordm*tea, 
and we already know that the factor Vg . d^ is invariant. 



THE PRINCIPLE OF LEAST AOTIOX. 75 

matter and radiation in bulk we may add a fourth, term 
involving the entropy, so as to bring in thermodynamical 
phenomena, and so on. The variations are taken with respect 
to these parameters, their values at the boundary of integration 
being kept constant. 

It is well known that the laws of mechanics of matter and 
of electrodynamics can be expressed in this form, so that we 
are here chiefly concerned with H. We already know from 
(424) that Einstein's theory is given by H-^G. Now G is, 
in fact, the principle invariant of the quadratic form g^dx^dx,,, 
viz., the Gaussian invariant of curvature. This aspect of the 
theory seems to eliminate any element of arbitrariness which 
may have been felt when we fixed on the contracted Biemann- 
Christoffel tensor for the law of gravitation. 

To interpret G as a curvature, consider a surface drawn in 
space of five dimensions, whose equation referred to the lines 
of curvature and the normal (z) at a point on it may be written 

2z=Jc l x^ + k^xl+k 3 xl+k^+'big^i powers . (44-2) 

where k l9 Jc& & 3 , & 4 are the reciprocals of the principle radii of 
curvature. 

Then ds* =dz*+ Zdx\. 

Eliminating % by (44-2) 

ds 2 ~(l+k\xl)dal+ ____ +2* 1 A; a a? 1 a? 2 diB 1 da? t +. . (44*3) 
Hence at the origin, 

9W=1, SW=0 (M~ v )> dgjdx<r=0. 
The only surviving terms in G=g f " v G^ v are 

-^ 
We easily find that 



In three dimensions we have only two curvatures, and i]& 2 
is known as Gauss's measure of curvature, i.e., the ratio of the 
solid angle contained by the normals round the perimeter of 
an element to the area of the element. The expression (44-4) 
is a generalisation of this invariant to five dimensions. 

The curvature G in ordinary matter is quite considerable. 
In water the curvature is the same as that of a spherical space 
of radius 570,000,000 km. Presumably, if a globe of water of 
this radius existed, there would not be room in space for 
anything else. 



76 RELATIVITY THEORY OF GRAVITATION. 

45. Electromagnetic Equations. 

The electromagnetic field is described by a covariant vector 
HL. In Galilean co-ordinates, 

X^PI-GIH, <D), . . . (45-i) 

where F, G, H is the vector potential and <D the scalar 
potential of the ordinary theory. 

If n^ is the covariant derivative of # M , we have by (22-2) 

J*>, =#^#^=8 covariant tensor, 

=F v , say (4:5*2) 

The electric and magnetic forces are given in the electro- 
magnetic theory by _ _ ^ ^ 



- - - 



Hence by (45-2) the value of -F^ in Galilean co-ordinates is 

= -y ^ -Z . . . (4541) 

y -a -F 
-ft a -Z 
v X Y Z 

and the associated contravariant tensor, F^=-g^g"^F^, is 

f~ = -y fi Z . . . (45-42) 

y -a F 
-fa Z 

_x -y -z o 

We can now express Maxwell's equations in covariant form. 
In the ordinary theory they are 

9 ?_?!_J^ 3XJ& = _W ZYJX^J^ _ 
dy 8z~~ & ' dz fa dt ' 3 9y ' ^ ' ' 

8a dZ 

(45>52) 



where the velocity of light is unity, and the Heaviside-Lorentz 
unit of charge is chosen so that the factor in disappears. The 



THE PRINCIPLE OP LEAST ACTION. 77 

electric current u, v, w and the density of electric charge p 
form a contravariant vector, since 

. fdx du dz dt\ . . * 

(u, v, w, p) = 2e(^ , , 5 , w ) per unit volume,* 

W% say .......... (45-6) 

Equations (45 51) and (45 54) may be written, 

3 ^ + ^- + ^ = 3 .... (45-71) 
dx* 3^ #* v ' 

and the remaining equations (45*52) and (45-53) give 



Now (45-71) is satisfied identically on substituting the values 
of F^ from (45 2), so that (45-2) and (45-72) represent the 
fundamental electromagnetic equations. The former is already 
co variant, and the latter Is made co variant by writing the 
co variant derivative ior the ordinary derivative. Thus 



(45-81) 

(45-82) 

v ; 



are the required equations. These hold in the gravitational 
field because the conditions for the application o the principle 
of equivalence ( 27) are satisfied. 

The expression Fy may be simplified as in 39 ; but owing to 
the antisymmetry of F^ the term corresponding to (39-3) dis- 
appears, and the equation reduces to 

" (45-9) 

^ 



-g 3 

The fact that Maxwell's equations can be reduced to a co- 
variant form shows that all electromagnetic phenomena 
described by them will be in agreement with the principle of 
relativity. 

* The occurrence of d$ in#bead of At in the denominator is due to the 
Michelson-Morley contraction, $=dt/d$, which makes the estimate ol tuut 
volume by a fixed observer dmer from that made by an observer moving 
with the electrons. (Gf. equation '7 $5).) 

G 



78 RELATIVITY THEORY OF GRAVITATION. 

46. The Electromagnetic Energy-Tensor. 

According to the electromagnetic theory, the components of 
mechanical force on unit volume containing electric charges 
are 



k 2 ~ p F-j- aw y 
& 8 =pZ+$w a 

and the negative rate of doing work is 



since the magnetic force does no work. 
By (4541) and (45-6), these give 



so that k v is a vector. 

But k v represents the rate at which the momentum and 
negative energy of the material system are being increased, 
i.e., in Galilean co-ordinates, 



If there exists a corresponding tensor Z?Jf or the electromagnetic 
field, this must change by an equivalent amount in the opposite 
direction in order to satisfy the law of conservation. Tbus 



It is not difficult to show from (46-1) and (46-3) that 

Et=-F vft F'* + ig*F'"Fn. . . . (464) 

We omit the proof as the precise value is not of great interest 
to us. It is sufficient to know that the expression is of the 
necessary tensor-form, so that an energy-tensor for the electro- 
magnetic field exists 

In general co-ordinates (46 2) and (6-3) are replaced by 
the covanant equations, 

-T; a =t,=ff; a . ... (405) 
in accordance with the principle of equivalence. 

When no matter is present this gives $" a =0, and we can 
derive the reaction of the gravitational field just as in (39-5). 
It follows that electromagnetic energy in the gravitational 
field experiences a force just as material energy does. Further 



THE PRINCIPLE OF LEAST ACTION. 79 

lectio magnetic energy exerts gravitation, because (39-13) and 

(46 5) give 



the lower a denoting covariant differentiation. 
Hence on integrating, (3942) must be replaced by 



In fact the electromagnetic energy-tensor must simply be 
added on tcrthe material energy-tensor throughout our work. 
When V g=l, we have the most general law of conserva- 
tion for triangular interchanges between matter, electro- 

magnetism and gravitation. 



^O- . . . . (46-6) 

a 

47, The Aether. 

The application of the Calculus of Variations to (44*1) gives 
a number of differential equations equal to the number of 
parameters varied ; but, according to a general theorem due 
to Hilbert, there are always four identical relations between 
these equations (the number 4 corresponding to the dimensions 
ot dt). The number of independent equations is thus four 
less than the number of unknowns, so that in addition to 
arbitrary boundary conditions we can impose four arbitrary 
relations on the parameters. It is this freedom of choice of 
co-ordinates that is so fundamental a characteristic of the 
generalised principle of relativity 

If we vary H l only we find the ten equations (3^=0. The 
identical relations in this case have been given in 39. If we 
vary the electromagnetic variable K^ as well, we get 14 
equations, of which 10 are independent, to determine 14 un- 
kno wns Within certain limits we can give arbitrary values to 
four of the unknowns, and the other ten willthen be determined 
definitely by the equations and the boundary conditions. If 
we elect to fix the values of the four co-ordinates A in this 
way (so that they are, as it were, disposed of) the g^ will 
become fixed, that is to say, there will be only one possible 
space-time. The phenomena, electromagnetic as well as 
gravitational, will all be described by the g^ which represent 
the state of strain of this space-time. This space-time mav 
be materialised as the aether, and the aether-theory does in 
fact attribute electromagnetic phenomena to strains in this 
supposed absolute medium. 

G 2 



RELATIVITY THEORY OF GRAVITATION. 

This is only a crude indication of the relation of the aether- 
theory to our relativity theory. As is well known, the modern 
aether-theory involves rotational strains. Moreover, we can- 
not get rid of the electromagnetic variables by putting them 
equal to zero, because they form a vector, which cannot vanish 
in one system of co-ordinates without vanishing in all. 

48. Summary of the Last Two Chapters. It may be useful 
to review the results which have been obtained from the point 
at which we introduced the energy-tensor T^ of the material 
system. Initially it was brought in for the practical purpose 
of calculating the gravitational field of a material body ; but 
this has led on to a discussion of the general laws of dynamics. 

As mentioned in 6, it is important, if we wish to adopt the 
principle of relativity, to show that the laws of nature which 
we generally accept are consistent with the principle ; or ? if 
not., to modify them so that they may become consistent. We 
have had to modify one law the law of gravitation. The 
laws of mechanics (Newton's laws of motion) are equivalent 
to the conservation of momentum and the conservation of 
mass. We have in 7(c) found it necessary to generalise the 
latter by admitting that energy has mass , and the conservation 
of mass is absorbed in the conservation of energy. The most 
generabstatement of these two principles of conservation for 
material systems is found in the general equations of hydro- 
dynamics (or of the theory of gases), viz , (37-7) and (37-8), 
and it is therefore sufficient to verify these. We have done 
that by showing that they may be expressed in tensor-form. 
We have even gone further ; we have shown that these laws 
can actually be deduced from the law of gravitation. They 
correspond to the four identical relations between Einstein's 
ten equations of gravitation ( 39). 

It has similarly been verified that our electromagnetic 
equations are of tensor-form and are therefore consistent 
with relativity. But in this case we have not deduced the 
electromagnetic equations from anything else ; we have 
merely shown their admissibility. The energy-tensor JE of 
the electromagnetic field is found from the consideration that 
in interchanges between the material and electromagnetic 
systems the total momentum and energy must remain constant. 

When the co-ordinates are not Galilean, gravitational forces 
will be acting and the total energy and momentum of the 
material and electromagnetic systems will be altering. We 



THE PRINCIPLE OF LEAST ACTION. 81 

have shown how to find this flux of energy and momentum 
(39-5), and in 43 we have traced it into the gravitational 
field, showing that it reappears there as the quantity , which, 
moreover, is conserved when no transfer of this kind is going 
on. There is, however, one reservation necessary ; unlike 1^ 
and E* 9 t% is not a tensor, and in order that this complete 
conservation of energy and momentum may be apparent we 
have to choose co-ordinates so that V - g=l. This does not 
imply any exception to the physical law of conservation, 
because we can always choose co-ordinates satisfying this 
condition It is merely that the energy-tensor is slightly 
more general than the physical idea of energy and momentum ; 
the former may be reckoned with respect to any co-ordinates, 
the latter must be reckoned with respect to co-ordinates 
satisfying V g=l. 

From the existence of an energy-tensor for the electro- 
magnetic field, it is deduced that electromagnetic energy must 
experience and exert gravitational force. 

The remainder of our work has been principally concerned 
with showing that our equations are equivalent to a principle 
of least action. From a theoretical standpoint there is a great 
deal to be said in favour of reversing the whole procedure, 
starting from the principle of least action as a postulate ; but 
I have preferred the present course as more elementary. 

Some difficulty may be found in the fact that the time- 
component of a four-dimensional vector is usually called by a 
different name from the space-components. The following 
table may be useful for reference : 

Vector. Space-Components. Time- Component. 

Ti negative momentum energy (mass). 

T l flux of negative momentum flux of energy (mass). 

kf,. ...... force negative rate of doing work. 

KH negative vector potential... electric scalar potential. 

JA ...... electric current-. density" ... electric charge-density. 



CHAPTER VIII. 



THE CURVATURE OF SPACE AND TIME. 

49. We have now presented the laws of gravitation, of 
hydromechanics, and of electro magnetism, in a form which 
regards all systems of co-ordinates as on an equal footing. 
And yet it is scarcely true to say that all systems are equally 
fundamental ; at least we can discriminate between them in a 
way which the restricted principle of relativity would not 
tolerate. 

Imagine the earth to be covered with impervious cloud. By 
the gyro-compass we can find two spots on it called the Poles, 
and by Eoucault's pendulum-experiment we can determine an 
angular velocity about the axis through the Poles, which is 
usually called the earth's absolute rotation. The name fck abso- 
lute rotation " may be criticised ; but, at any rate, it is a name 
given to something which can be accurately measured On 
the other hand, we fail completely m any attempt to determine 
a corresponding * k absolute translation " of the earth It is 
not a question of applying the right name there is no measured 
quantity to name. It is clear that the equivalence of systems 
of axes in relative rotation is in some way less complete than 
the equivalence of axes having different translations ; and 
this may perhaps be regaided as a failure to reach the ideals 
of a philosophical principle of relativity. 

This limitation has its practical aspect. We might suppose 
that from the expression (28*8) for the field of a particle at 
rest it would be possible by a transformation of co-ordinates 
to deduce the field of a particle, say, in unifoim circular motion. 
But this is not the case. We may, of course, reduce the 
particle to rest by using rotating axes ; but we find it necessary 
to take an entirely different solution of the partial differential 
equations, satisfying different boundary conditions. 

We have not hitherto paid any attention to the invanance 
of the boundary conditions ; and it is here that the break- 
down occurs. The ases ordinarily used in dynamics are suck 



THE CURVATURE OF SPACE AND TIME. 83 

that as we recede towards infinity in space the g^ approach 
the special set of values (16-3). On transforming to other co- 
ordinates the differential equations are unaltered ; but usually 
the boundary values of the g^ v , and consequently the appro* 
priate solutions of the equations, are altered. We can, there- 
fore, discriminate between different systems of co-ordinates 
according to the boundary values of the #'s ; and those which 
at infinity pass into Galilean co-ordinates may properly be 
considered the most fundamental, since the boundary values 
are most simple. The complete relativity for uniform trans- 
lation is due to the boundary values as well as the differential 
equations remaining unaltered.* 

We have based our theory on two axioms the restricted 
principle of relativity and the principle of equivalence. These 
taken together maybe called the physical principle of relativity. 
We have justified, or explained, them by reference to a philo- 
sophical principle of relativity, which asserts that experience 
is concerned only with the relations of objects to one another 
and to the observer and not to the fictitious space-time frame- 
work in which we instinctively locate them. We aie now led 
into a dilemma ; we can save this philosophical principle only 
by undermining its practical application. The measurement 
of the rotation of the earth detects something of the nature 
of a fundamental frame of reference at least in the part of 
space accessible to observation. We shall call this the 
" inertial frame." Its existence does not necessarily contradict 
the philosophical principle, because it may, for instance, be 
determined by the general distribution of matter in tiie 
universe ; that is to say, we may be detecting by our experi- 
ments relations to matter not generally recognised. But 
having recognised the existence of the inertial frame, the 
philosophical principle of relativity becomes arbitrary in its 
application. It cannot foretell that the Michelson-Morley ex- 
periment will fail to detect uniform motion relative to this 
frame ; nor does it explain why the acceleration of the earth 
relative to this frame is irrelevant, but the rotation of the 
earth is important. 

The inertial-frame may be attributed (1) to unobserved 
world-matter, (2) to the aether, (3) to some absolute character 

* Owing to the four additional conditions that can be imposed on the (fn 
the boundary values are not sufficient to determine the co-ordinates uniquely 
and the principle of relativity is valid in its most complete sense for trans- 
formations considerably more general than uniform translation. 



84 RELATIVITY THEORY OF GRAVITATION. 

of space-time. It is doubtful whether the discrimination 
between these alternatives is more than word-splitting, but 
they lead to rather different points oi view. The last alterna- 
tive seems to contradict the philosophical principle of relativity, 
but in the light of what has been said the physicist has no 
particular interest in preserving the philosophical principle. 
In this chapter we shall consider two suggestions towards a 
theory of the inertial frame made by Einstein and de Sitter 
respectively. These should be regarded as independent specu- 
lations, arising out of, but not required by, the theory hitherto 
described. 

The inertial frame is distinguished by the property that the 
g^ referred to it approach the limiting Galilean values (16*3) 
as we recede to a great distance from all attracting matter. 
This is verified experimentally with considerable accuracy ; 
but it does not follow that we can extrapolate to distances 
as yet unpluinbed, or to infinity. If it is assumed that the 
Gf-alilean values still hold at infinite distances, the inertial 
frame is virtually ascribed to conditions at infinity, and its 
explanation is removed beyond the scope of physical theory. 
We may, however, suppose that observational results relate to 
only a minute part of the whole world, and that at vaster 
distances the g^ tend to zero values which would be invariant 
for all finite transformations. In that case all frames of 
reference are alike at infinity, and the property of the inertial 
frame arises from conditions within a finite distance. In that 
case physical theories of the inertial frame may be developed. 

The ascription of the inertial frame to boundary conditions 
at infinity may also be avoided by abolishing the boundary. 
This is really only another aspect of the vanishing of the g hv 
at infinity. Our four-dimensional ^pace-time may be regarded 
as a closed surface in a five-dimensional continuum ; it will 
then be unbounded but finite, just as the surface of a sphere 
is unbounded. 

We have seen (44) that wherever matter exists space-time 
has a curvature. It might seem that if there were sufficient 
matter the continuum would curve round until it closed up ; 
but it has not been found possible to eliminate the boundary 
so simply. I think the difficulty arises because time is not 
symmetrical with respect to the other co-ordinates ; in general 
matter moves with small velocity, so that the different com- 
ponents of the energy-tensor T% are not of the same order of 
magnitude. 



THE CURVATURE OF SPACE AND TIME. 85 

50. Einstein suggests that in measurements on a vast scale 
the line-element has the form 



This expression includes the effects of the general distribution 
of matter through space ; but there will be superposed the 
local irregularities due to its condensation into stellar systems, 
etc. 

The expression (50-1) can be interpreted* as belonging to a 
three-dimensional space which forms the surface of a hyper- 
sphere of radius jR in four dimensions, the time being recti- 
linear. Let be the origin of co-ordinates (Kg. 5), A the 




FIG. 5. 



centre of the hypersphere, and % the angle OA C. If is the 
azimuthal angle of the plane OAC, the line-element at C for 
an ordinary sphere would be 



The expression (50-1) is the extension of this for an extra 
dimension measured by 9. 

In the figure the circumference of the circle CC' is 2rcjRsin%, 
but its radius measured along the spheie is R%. Similarly in 
our curved space the surface of a sphere of radius R% will be 
; successively more distant spheres will increase in 



* Other interpretations are possible ; but this is probably the easiest 
conception for those unfamiliar with non- Euclidean geometry. For this 
reason I do not here describe the interpretation in terms of " elliptical space, 
which has certain advantages. 



86 RELATIVITY THEORY OF GRAVITATION. 

area up to a radius %nR, and afterwards decrease to a point 
for the limiting distance nR. The whole volume of space is 
finite and equal to 2?i 2 R* in natural measure.* 

From (50*1) the values of G> can be calculated just as in 
28. We find, in fact, 

G^jza,** except 044=0 1 

; . . . (50-2) 

so that = R* ' 

Hence by (35 8) 

1 3 

=-~^i except -8^44== - 2 . . (503) 



Unless we are willing to suppose that the matter in the universe 
is moving with speeds approaching that of light, T^ is much 
greater than the other components, and it is clearly impossible 
to satisfy (50-3). The only possible course is to make a slight 
modification of the law of gravitation. Neglecting the motion 
of matter we shall have T M = p, and the other components 
vanish. The modified law that satisfies (50 2) must then be 

. . . (504) 



where 1=1 /R* and p^ 

Equation (504) replaces (35*8). The radius R may ^ be as 
great as we please, so that we may satisfy our scruples without 
introducing any modification perceptible to observation. 

In Hamilton's principle becomes replaced by 041, 
and space-time has a natural curvature 41 when no matter is 
present ; this curvature is increased to 6/1 where there is 
matter having the average density. (Cf. (444) with & 4 =0.) 

Since the whole volume of space in natural measure is 
27r 2 JS 3 5 the total mass of matter is 2jt*R*p=%aR. The mass 
of the sun is 1 47 kilometres ; the mass of the stellar system may 
be estimated at 10 9 Xsun; let us suppose further that the 
spiral nebulae represent 1,000,000 stellar systems havmg this 
mass. Even this total mass will only give us a universe of 
radius 10 16 kilometres, or about 30 parsecs much less than 
the average distance of the naked-eye stars. Einstein's 
hypothesis therefore demands the existence of vast quantities 
of undetected matter which we may call world-matter. 

* The observer will probably Introduce measures more convenient to 
himself (c/. 52), so that in his co-ordinates the limiting distance may be oo 
or even beyond. 



THE CURVATURE OP SPACE AOT) TIMS, 87 

Some curious results are obtained by fol >wing out the 
properties of this spherical space. The j^arallax of a star 
diminishes to zero as the distance (in natural measure) increases 
up to ^TiR ; it then becomes negative and reaches 90 at a 
distance nR. Apart from absorption of light in space we 
should see an anti-sun, at the point of the sky opposite to the 
sun equally large and equally bright,* the surface-markings 
corresponding to the back of the sun. After travelling " round 
the world " the sun's rays come back to a focus. Since p and 
R are Delated, it has been suggested that we can use the 
invisibility of this anti-sun to give a lower limit to R, assuming 
that no light is lost in space except by the scattering action 
of the world-matter. But it appears to have been overlooked 
that Einstein's new hypothesis is inconsistent with relativity 
in its ordinary sense ; the anti-sun will not be a virtual image 
of the sun as it is now, but of the sun as it was when it emitted 
the light perhaps millions of years ago, when it was in another 
part of the stellar system. Einstein has restored the diffe;- 
entiation between space and time by assuming the space-tin e 
world to be cylindrical, so that the linear direction gives an 
absolute time. It is only locally that we can still mate 
Minkowski's transformation ; rigorously the physical punciple 
of relativity is violated since space-tune is no longer isotropic. 

We regret being unable to recommend this rather picturesque 
theory of anti-suns and anti-stars. It suggests that only a 
certain proportion of the visible stars are material bodies, 
the remainder are ghosts of stars, haunting the places where 
stars used to be in a far-off past. 

Owing to this violation of the restricted principle of relativity 
we have a feeling that Einstein's new hypothesis throws away 
the substance for the shadow. It is also open to the serious 
criticism that the law of gravitation is made to involve a 
constant 1, which depends on the total amount of matter in 
the universe (A=7i a /4M 2 ). This seems scarce y conceivable ; 
and it looks as though the solution involves a" very artificial 
adjustment, 

51. An alternative proposal has been made by de Sitter 
which seems much less open to objection. He takes for the 
line element 



(5ia) 

For constant time the three-dimensional space is spherical as- 
in (50-1) ; but there is also a curvature in the time- variable. 

* Disregarding the sun's absolute motion referred to later. 



88 RELATIVITY THEORY OF (3-RAVITATION. 

With the present variables this is not of a simple kind, but 
setting 

sin%=smf sin o> j , . (51-2) 
tan (it/R)=eo$ Ctan oJ 
we find 



which corresponds to spherical polar co-ordinates (jR, a>, f, 0, 9) 
in space of five dimensions. By measuring C from different 
azimuths we perform an operation corresponding to Min- 
kowski's rotation of the time-axis, so that there is here DO 
absolute time, and the original principle of relativity is fully 
satisfied. 

The properties of de Sitter's space-time are best recognised 
from (51-1). Near the origin we have ordinary Galilean space- 
time. As we recede, space has the spherical properties already 
mentioned, and in addition measured time (ds) begins to run 
slow relati ve to co-ordinate time (dt). Fin allv at v &fc a ^#. at 
A natural distance f TtR, time stands still. At any fixed point 
ds is zero however large dt may be, so that nothing whatever 
can happen however long we wait. 

Of course, this is merely the point of view of the observer 
at the origin of co-ordinates. All parts of this spherical con- 
tinuum are interchangeable ; and if our observer could transport 
himself to this peaceful abode, he would find Nature there as 
active as ever. Moreover, adopting the co-ordinates natural 
to his new position, he would judge his old home to be in this 
passive state. There is a complete lack of correspondence 
between the times at the two places. They are, as it were, 
at right angles, so that the progress of time at one point has 
no relation to the perception of time at the other poiat 

The line-element (51-1) leads to 



and accordingly the law of gravitation is taken to be (504), 
with 

A=3/J2 2 . 

The aggregate curvature due to matter is here neglected in 
comparison with the natural curvature due to the modification 
of the law of gravitation, and there is no assumption of the 
existence of vast quantities of matter not yet recognised. 



THE CURVATURE OF SPACE AlSiB TIME. 89 

There is no anti-sun on de Sitter's hypothesis,, because light, 
like everything else, is reduced to rest at the zone where time 
stands still, and it can never get round the world. The region 
beyond the distance ^nR is altogether shut off from us by 
this barrier of time. The parallax of a star at this distance 
will be such as corresponds to a distance R in Euclidean space, 
and this is the minimum value possible. 

The most interesting application of this hypothesis is in 
connection with the very large observed velocities of spiral 
nebulae, which are believed to be distant sidereal systems. 
Since <\/g^=(x>s %, the vibrations of the atoms become slower 
(in the observer's time) as cos % diminishes, in accordance 
with 34. We should thus expect the spectral lines to be 
displaced towards the red in very distant objects, an effect 
which would in practice be attributed to a great velocity of 
recession. It is not possible to say as yet whether the spiral 
nebulae show a systematic recession, but so far as determined 
up to the present receding nebulae seem to preponderate. 

Superposed on the (spurious) systematic radial velocity will 
be the individual velocities of the nebulae. It is scarcely 
possible to say what these are likely to be without making 
some assumption. There is no meaning in absolute motion, 
and if two systems are entirely independent, so that their 
relative motion has no physical cause, it must be quite arbitrary, 
and there is no reason to expect it to be small compared with 
tho velocity of light. If, however, the systems have separated 
from one another, it can be shown by rather laborious calcu- 
lations* that their velocities will tend to become more diverse 
as they recede, up to the limit ^nR for which the velocities 
are comparable with that of light. We should thus have an 
explanation of the large velocities of the spirals, averaging 
300-400 km. per sec., and we could perhaps form an estimate 
of the value of R. 

It must be remembered that in natural measure the internal 
motions of stars in a spiral system will be of the same magni- 
tude as in our own system, owing to the homogeneous character 
of de Sitter's space-time. In co-ordinate measure these in- 
ternal motions will be smaller owing to the transformation of 
the time. The possibility of large divergent motions of the 
systems as a whole depends on the large separation between 
them. 

* De Sitter, " Monthly Notices," November, 1917. 



90 RELATIVITY THEORY OF GRAVITATION, 

52. So far we have used spherical co-ordinates, but we can 
map the spherical space oi Einstein or of de Sitter on a fiat 
space by performing the central projection r=R tan % 
r will be"represented by OP in Fig. 5, and the variables r, 6, c* 
will satisfy Euclidean geometry. This does not mean that 
measured space is Euclidean ; but that we multiply our 
measures by suitable factors in order to obtain results which 
will fit together in Euclidean space, just as we did for a local 
gravitational field in 28. With r as variable (50 I) and 
(51-1) become, respectively, 



where sl/R 2 . 

These show that at *' infinity " (i.e., r=<x> ) the values oi g^ 
in rectangular co-ordinates approach the respective limits. 

EINSTEIN. DE SITTER. GALILEO. 

0000 0000 -1000 

0000 0000 0-100 

0000 0000 00-10 

0001 0000 0001 

the Galilean values being added for comparison. 

De Sitter's limiting values are invariant for all transforma- 
tions ; Einstein's only for transformations not involving the 
time ; the Galilean values for the transformation oi uniform 
motion and a limited group of other transformations 

De Sitter's hypothesis thus appears to present the greatest 
advantages ; but it will not satisfy the followers of Mach's 
philosophy. He derives his mertial-f rame from the spherical 
property of space-time which in turn is derived from the slightly 
modified law of gravitation ; it is not determined by anything 
material The followers of Mach maintain that if there were 
no matter there could be no inertial frame, and it appears 
that this is Einstein's reason for preferring his own suggestion. 
In his theory if all matter were abolished, R would become 
zero and the world would vanish to a point. There is some- 
thing rather fascinating in a theory of space by which, the 
more matter there is, the more room is provided. It is 
satisfactory, too, from Einstein's standpoint, because he is 
unwilling to admit that a thinkable space without matter 



THE CURVATURE OP SPACE AND TIME. 91 

could exist. For our pait, we feel equally unwilling to assent 
to the introduction of vast quantities of world-matter, which, 
(to quote de Sitter) " fulfils no other purpose than to enable 
us to suppose it not to exist." 

53. In this discussion of the law of gravitation, we have 
not sought, and we have not reached, any ultimate explanation 
of its cause. A certain connection between the gravitational 
field and the measurement of space has been postulated, but 
this throws light rather on the nature of our measurements 
than on gravitation itself The relativity theory is indifferent 
to hypotheses as to the nature of gravitation, just as it is 
indifferent to hypotheses as to matter and light. We do not 
in these days seek to explain the behaviour of natural forces in 
terms of a mechanical model having the familiar characteristics 
of matter in bulk ; we have to accept some mathematical 
expression as an axiomatic property which cannot be further 
analysed. But I do not think we have reached this stage in 
the case of gravitation 

There are three fundamental constants of nature which stand 
out pre- eminent K T 

The velocity of light, 3 00. 10 10 C.GLS units ; dimensions LT~\ 
The quantum, 6-55. 10' 27 , ML*T~\ 

The constant of gravitation, 6 66 . 1Q~ 8 ; M- 1 L*T-*. 

From these we can construct a fundamental unit of length 
whose value is 

4XlO- 33 cms. 

There are other natural units of length the radii of the 
positive and negative unit electric charges but these are of 
an altogether higher order of magnitude. 

With the possible exception of OsborneReynolds's theory of 
matter, no theory has attempted to reach such fine-grainedness 
But it ts evident that this length must be the key to some 
essential structure. It may not be an unattainable hope that 
some day a clearer knowledge of the processes of gravitation 
may be reached ; and the extreme generality and detachment 
of the relativity theory may be illuminated by the particular 
study of a precise mechanism. 



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